In a liquid measuring cup or small bowl combine eggs, salt, and black pepper. Beat with a fork until combined, but not frothy; set aside.
In a 10-inch nonstick skillet with flared sides heat 1 tablespoon of the butter over medium heat until melted. Add green and red sweet pepper and onion; cook about 3 minutes or until vegetables are tender, stirring occasionally. Remove vegetables from skillet; keep warm.
For omelet, in the same skillet heat 1 1/2 teaspoons of the remaining butter over medium-high heat until melted. Pour half of the egg mixture (about 1/2 cup) into skillet; stir in one-fourth of the cooked vegetables. Cook over medium heat. As mixture sets, run a spatula around edges of skillet, lifting egg mixture so the uncooked portion flows underneath. Continue cooking and lifting edges until egg mixture is set but is still shiny.
Arrange half of the ham, half of the avocado, and half of the cheese across center of omelet. Using the spatula, lift and fold one edge of the omelet about a third of the way toward center. Fold the opposite edge toward center. Gently slide omelet out of skillet onto a warm plate. Cover with foil; keep warm. Repeat to make a second omelet, using the remaining 1 1/2 teaspoons butter, one-fourth of the cooked vegetables, and the remaining egg mixture; fill with the remaining ham, avocado, and cheese. Spoon the remaining cooked vegetables over omelets before serving.