Beat the eggs and half-n-half together in a bowl, set aside
In a saute pan, melt the butter over medium heat. Add the ham and vegetables and warm them throughout.
Pour in the egg mixture, and spread evenly throughout pan.
Cook until the edges have little moisture (there can still be some moisture in the middle. Flip the omelet over. You can use a big spatula, or the wrist flip if you're feeling daring.
Sprinkle the cheese on the omelet. Cook for about 1 minute, and then fold over in half, and then slide onto a plate to serve.