Unless I'm misunderstanding you, we don't use "to" the way you say... but I'm basing this on American English. Other varieties of English may use the prepositions differently.
To my understanding, this sentence means that cook something under the right temperature. Or that you need to know the right temperature of what you are cooking and cook it under that temperature.
I'll try to explain the usage of "cook to" in a way that should help
"Cook to" is usually seen in recipes when you're being told to cook something until it reaches a specific temperature or to bring the food's temperature up to a particular point:
Rack of Lamb: Rack of lamb should be cooked at 325 degrees. For rare; cook approximately 1 hour or to an internal temperature of 130 degrees. For medium rare; cook to an internal temperature of 140 degrees. [emphasis added]
In this example, the internal temperature is what tells you that the lamb has reached the preferred level of doneness and even the amount of time it cooks and the temperature of the oven can't guarantee that it will have reached that internal temperature, so it is necessary to use a thermometer.
Here's a great image of a rack of lamb with a meat thermometer in it... as you can see in the image, the internal temperature is just above 130 F.
Image from here.
This is likely using definition 10 of "to" here:
10 a) as far as a particular point or limit:
- Temperatures dropped to 25 degrees below zero.
As someone who cooks and reads a lot of recipes, this is the only definition of "to" that works here. Even metaphoric phrases make the same definition of "to":
The steak was cooked to death. (the steak was really overcooked)
The steak was cooked to perfection. (the steak was cooked perfectly)
In fact, the other way that this is phrased is
Cook until it reaches an internal temperature of 140 degrees.
When you say something should be cooked "under" the right temperature, I think you are correct to believe it should be "at" or "in". Let's go back to the example I posted above but look at a different sentence this time:
Rack of Lamb: Rack of lamb should be cooked at 325 degrees. For rare; cook approximately 1 hour or to an internal temperature of 130 degrees. For medium rare; cook to an internal temperature of 140 degrees. [emphasis added]
So, in this sentence, when it's being explained what temperature you need to set your oven for, we use "at".
This happens to also be definition 10 for the preposition "at":
10 used to show a price, rate, level, age, speed etc:
- The Renault was traveling at about 50 mph.
I think of it as "the oven is set at 325 degrees".
With slight rephrasing, we also often use "in":
Rack of lamb should be cooked in a 325 degree oven.
And for this one, you can think of the fact that you're putting the food inside an oven that is 325 degrees.
As to "under", there's only one example I can think of where "under" would be appropriate instead of "at" or "in" and that's in the case of using a broiler.
A broiler is a special heating element in an oven that is designed to cook food from above. Placement of heating units in ovens varies by oven but broilers are always above the food when actively broiling.
In the United States, when the heat source for grilling comes from above, grilling is termed broiling.
Because of this, we regularly say "under the broiler".
- Spray a baking sheet with cooking spray (or coat lightly with olive oil) to prevent sticking. Lay the chicken breasts side by side not allowing them to touch each other. Cook under the broiler for about 5 minutes on each side, or until slightly charred and cooked through. Remove the chicken from the oven, baste with barbecue sauce, and broil for another minute. Remove from the oven and serve.
Yes, it both "sounds wrong" to a native speaker, and it doesn't make sense.
The phrase "[person] has been to [place]" simply means the person was present in the place at least once. It contains no information or implications about when they were there, for how long, or why. "has/have been to" is completely without duration. Asking "how long" makes no sense.
The question "Have you been to [place]?" is a question with a "yes" or "no" answer. Are you asking how long ago the person visited? Are you asking how long they stayed in the place? In either case, one would ask those questions instead.
Best Answer
There are several questions packed in here:
1) How much notice would a moment have? There is some reason that the phrase evolved to "moment's notice" rather than "a moment OF notice", but I was unable to find the history of the phrase to further clarify. Looking at it practically, I would not think that a moment would have very much notice. Further, since being poetic is giving words life beyond their obvious meaning, this is definitely poetic.
2) How did something poetic become common usage? Given that the use of the phrase seems to really catch on after 1791, I would assume that something that was popular used the phrase. Some gentle searching of famous works at that time (particularly poetry) did not reveal use of the phrase, so at this point anything I say would just be speculation.
3)a) (implied question) Is the possessive correct? It is but not for the reason you think. My research suggests that 24 hours' notice is incorrect, but moment's notice is correct. As you can see in the graph, "24 hours notice" is the older phrase and given that they both persist (despite hours' being currently and generally more popular), I usually go with the oldest being the winner. Further, the interpretation that "24 hours" is a measure of the notice seems reasonable. This also, I think, proves that this question is not a duplicate given that the phrases are different. The difference between the two classes of phrase is clear given the usage pattern moment's notice versus moments notice. Finally, it's clear that possessives are extremely complicated, so I would think that explains the lack of definitive discussion.
4) (Implied question) What about the beginning of the phrase? If you accept the usage data, the correct phrase is "at a moment's notice", at least according to the ngram viewer. I don't have a great answer for why though.
5) (Implied question) Is the moment personified? Yes. Unless you only use a narrow definition of personified. If you don't consider the relationship between the moment and the notice, the phrase makes no sense. Therefore, we must consider how much notice a moment would give, and to do that, you have to think about a moment having attributes like patience and forethought, which is very person-y. Ironically, I think the problem came when @FumbleFingers "misparsed" @RocketBouchard's sentence "The moment is personified and it is noticing." I think that the OP meant to use "noticing" as a verb as in "providing notice". Probably not the best choice when asking this kind of question. It also should be clarified that this personification is only happening with specific phrases like "moment's notice". Car's window, for instance, doesn't require us to know (or imagine) anything about the car or how it relates to the window.