What method of extracting oil from almonds preserves the most almond flavor

I think on some level you'll just get what you get with something hard like an almond. A really really sharp knife with a thin blade can help, but you can only do so much. My hand-cut ones are always a bit on the raggedy side.

When using a food processor, try smaller batches to leave bigger or more consistent pieces. You don't have to pulse as long to get everything broken up, so you get less that's ground to nothing. Do a small batch for a short chop, dump 'em out, do another.

As to how to do slivered or sliced almonds, I always figured that they started with blanched almonds (which are less crisp in my experience) or maybe even raw ones and then roasted them after cutting to crisp them up. You might experiment with blanched or unroasted almonds and see if you get a better result with a knife.

Why Alcohol?

Alcohol is used for extracts because the flavor compounds (plant oils) you are trying to extract do not easily dissolve in water. Alcohol (typically bourbon or vodka) will do the trick. Make sure you use +80 proof because it also acts as a preservative.

Making Mint Extract

To make an extract, tear up or coarsely chop and bruise washed mint leaves into a measuring cup (you'll end up with about twice the volume of extract as you have leaves). Transfer the leaves to a glass container with a tight-fitting lid. Add about twice the alcohol (by volume) as you had leaves. Cover and shake.

The mint leaves will tend to float to the top, but give it a chance. After a few days, they'll start to bog down with alcohol and sink. Shake it every few days or so. After a month, you'll have mint extract. Strain the leaves and store.

Adjusting the Strength of Your Extract

The longer you let the leaves steep, the stronger the extract will become until all the oils are essentially spent. You can sample the extract along the way until you get something to your liking. If you want something stronger, you can add fresh leaves to your strained extract and continue the process. There's a limit though; as the alcohol become saturated, you'll get diminishing returns by adding more leaves.

Freezing the extract will not congeal the oils for further separation. They're essentially dissolved in the alcohol (unlike water + oil) and the alcohol will not freeze. Extracts are typically too concentrated to drink straight. For all that effort, it's better just to crush a few leaves directly into a drink with whatever ingredients will make it a proper cocktail.

Boiled Leaves isn't Really Mint Extract

The boiling water method you mentioned above wont achieve the same results. Even if concentrated, the flavor compounds in extracts are typically somewhat volatile (which is why you add them near the end of cooking). You're basically making concentrated, flavored mint tea… but it isn't really an extract. And without the alcohol acting as a preservative, you're mint tea will have a somewhat limited shelf life. Even distilling the volatile oils by boiling and condensing into a liquid might get you pure mint oil, but that would likely need very specialized equipment.