Most creme brulees require baking, however after a little research I did find a recipe in "On Cooking" (Sarah Labensky/Michael Hause) that came from Chef Vincent Guerithault of Vincent on Camelback in Phoenix, AZ and his was similar in that it was not baked.
First, just making creme anglaise with heavy cream isn't going to do anything to let it set up into a firm custard. More egg yolks or starch would be needed.
Supposing that this really does work and it was something you perhaps did, my guess would be that it was either mixed too much (breaking down the proteins trying to link together) or too vigorously (incorporating air which weakened the protein links). In your description you say you "beat it". Did you beat it or stir it? It should be stirred back and forth zig-zagging across the bottom of the pan with a wooden spoon or heat-safe rubber spatula to keep from whipping air into it.
Time, temperature, and eggs/dairy ratio are going to be the main issues in getting custards to set.
Egg proteins begin to set at 160 degrees but curdle at 180 so there's very little "wiggle" room temperature wise.
According to Shirley Corriher's "Cookwise": 2 egg yolks will just barely thicken 1 cup of milk or cream. Her Creme Anglaise recipe uses 5 egg yolks to 1 cup of milk and 1/2 cup heavy cream which is more yolks and less liquid than Julia's and this isn't intended to set up. 1 teaspoon of starch isn't going to provide the thickening power that is needed, it's there to keep the yolks from curdling as easily.
The recipe I use and many others I've referenced (including Chef Vincent's), use a ratio of about 6-7 yolks per cup of cream.
Also, if using a starch, you need to nearly bring the custard mixture to a boil (as is common in puddings and cream pie fillings) otherwise an enzyme in egg yolks known as alpha-amylase will eat away at the starch bonds and break them down into a watery mess.
Chef Vincent's does not use any starch.
If you want to use that recipe, I would increase it to 10 egg yolks. After the hot cream is tempered into the egg yolks then return to the heat and cook, stirring constantly, until very thick but do not let it boil. Remove it from the heat and strain into a clean metal bowl and chill over an ice bath to cool quickly. Once cool, spoon into your desired serving dish or a cookie cup and caramelize the top with sugar.
You can use gelatin, but you then would have to change the process slightly. For a start, you wouldn't bake the custard. Instead you would essentially be making an egg-enriched panna cotta.
You would hydrate the gelatine with cold water, make your custard, then add the gelatine, mix and portion, then set in the fridge. How much gelatine you use depends on how much liquid you are trying to set - using powdered gelatin, one sachet usually sets 570ml.
Of course, that is cheating and you should really bake creme brulee with no gelatin. If you're having trouble getting regular creme brulee to set you may have a bad recipe - try another!
Best Answer
I see several problems here. First, the recipe only has 3 yolks per cup of liquid, that's the bare minimum for thickening and does not produce a really thick custard like the one expected in creme brulee. Second, it is a stirred custard, not a baked one, which is a method for producing pourable or at least creamy custards, not "spadeable" ones where you can take out a piece with your spoon without the sides slowly flowing to partially fill the hole. Third, it does not give you a temperature, but suggests you to use time and a visual clue ("coats metal spoon") and that clue is also consistent with a liquid custard like creme anglaise, not a thick one like creme brulee.
The best thing to do is to pick a different recipe for next time - one which uses more yolks and directs you to oven bake the custard in a water bath. Aim for 83-85 Celsius internal temperature (under 80 it will stay too soft, over 90 it will go grainy, so if you don't have a thermometer, it is very difficult to get it right). It will take several hours, that's normal.
For this batch, I wouldn't bother reprocessing it in any way. It is edible as it is, use it in whatever calls for a custard sauce. It's easiest and ensures that you won't run into additional trouble.
If you really insist on trying to make creme brulee out of this one batch, add more yolks and bake in the oven in a water bath until proper internal temperature.