Butter is an emulsion, a tenuous mixture of water and fat, and prone to separation when exposed to heat. This is a big deal when thickening garlic butter sauce. You can't bring butter sauce to a simmer like regular sauces, so gelatinizing starch thickeners take about twice as long as usual. You must also heat the sauce gently and add the thickening agent in the last minute or so of cooking it.
A liaison is perhaps the most elegant thickening agent, or at the least the one with the most finesse. A classic French thickener, a liaison is a mixture of egg yolks and cream that will thicken garlic butter sauce slightly without any starchiness or change in color. The sauce stays golden and smooth.
Mix 1 egg yolk and 2 ounces of heavy cream (for every 1 cup of sauce) in a measuring cup or mixing bowl until smooth and uniform in color. Add about 1/2 cup of hot garlic butter to the liaison and whisk to blend. Next, add the mixed liaison to the pan of garlic butter sauce. Cook the butter garlic sauce over low heat for four to five minutes, stirring frequently.
Garlic butter sauce thickened with flour essentially makes it a roux-thickened sauce. You typically have to simmer roux-thickened sauces for about 10 minutes to gelatinize the starch granules and produce a smooth texture. With garlic butter sauce, however, you have to gelatinize the starch over low heat for about twice as long, or until you can't detect any trace of grittiness.
For every 1 cup of garlic butter sauce, add 1 tablespoon of flour. Over low heat, whisk the sauce enthusiastically to incorporate the flour. Cook the sauce for 15 to 20 minutes or until it has a smooth texture when you taste it.
Cornstarch thickens relatively quickly, so you don't have the textural issues you can have with flour. But you have to dissolve cornstarch in water before adding it to the sauce and heat it for several minutes to cook off the water.
Mix 1 tablespoon of cornstarch in 1 tablespoon of cold water (for each cup of garlic butter sauce). Whisk the cornstarch into the butter. Cook the sauce over low heat until thickened, or about 5 to 10 minutes.
If you have a small quantity of garlic butter sauce, such as the amount used for one serving of shrimp scampi or pasta, sprinkle 1/2 teaspoon of flour over the surface and set the heat to low. Whisk the flour to blend and let it cook over low heat for three or four minutes before serving.