Omelets
Omelets are perhaps the best known egg dishes in North America. Beaten eggs are mixed with a little liquid (no more than 1 Tablespoon per egg, and often less), and cooked until set, then folded around a filling. They are usually eaten immediately after cooking.Tip: Make sure the filling is warm before putting it into the omelet.
For a step-by-step illustration of making a simple omelet, check out this Illustrated Omelet Guide by Fiona Haynes, Abouts Guide to Low Fat Cooking.
Frittatas
Lesser known in the United States, this is an Italian version of an omelet. Because of the way it is cooked, I think it is more versatile, as it can be easily eaten later, and even frozen. Several portions are usually cooked at once, in only marginally more time than it takes to cook an omelet. There are several techniques, but I favor a quick one that starts on the stove and finishes in a few minutes under the broiler. Note that while saucy fillings can work well in omelets, you would usually want to avoid putting sauces in a frittata.Tip: Small cubes of cheese in a frittata will melt during cooking and create yummy little cheese pockets.
How to Cook a Frittata
Recipe: Pizza Frittata
Quiches
A quiche is essentially a baked custard (savory rather than sweet) in a pie shell - although you can certainly make one without the crust, as I always do. It usually includes cheese, as well as other ingredients. Since it is a custard, it is more delicate in consistency than a frittata. This is because it is made with more liquid than eggs, traditionally 2 to 3 eggs per cup of liquid (traditionally cream, but this is less usual these days), although you see recipes with more eggs.Tip: The trick with quiche is how to preserve the delicate texture. This is achieved by removing it from the oven while it is still a bit uncooked in the center; it will continue to cook when removed from the heat. Overcooked quiche has a tough, cracked texture around the outside.
How to Make a Crustless Quiche
Recipe: Smoked Salmon, Leek, and Mushroom Quiche (Crustless)

