Melt one tablespoon butter in a medium skillet over medium heat. Place onion and bell pepper inside of the skillet. Cook for 4 to 5 minutes stirring occasionally until vegetables are just tender.
While the vegetables are cooking beat the eggs with the milk, 1/2 teaspoon salt and pepper.
Shred the cheese into a small bowl and set it aside.
Remove the vegetables from heat, transfer them to another bowl and sprinkle the remaining 1/4 teaspoon salt over them.
Melt the remaining 1 tablespoon butter (in the skillet just used to cook the vegetables) over medium heat. Coat the skillet with the butter. When the butter is bubbly add the egg mixture and cook the egg for 2 minutes or until the eggs begin to set on the bottom of the pan. Gently lift the edges of the omelet with a spatula to let the uncooked part of the eggs flow toward the edges and cook. Continue cooking for 2 to 3 minutes or until the center of the omelet starts to look dry.
Sprinkle the cheese over the omelet and spoon the vegetable mixture into the center of the omelet. Using a spatula gently fold one edge of the omelet over the vegetables. Let the omelet cook for another two minutes or until the cheese melts to your desired consistency. Slide the omelet out of the skillet and onto a plate. Cut in half and serve.
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