In a medium bowl, whisk together the eggs, milk (if using), salt, and pepper. Whisk in a side-to-side motion until uniform in appearance; do not over-mix. Set aside.

In an 8-inch nonstick skillet, melt 1 teaspoon of butter over medium heat until it just begins to bubble and foam, but does not brown. Add the diced ham and saute until lightly browned, 1 ½ to 2 minutes.

Add the diced onions, green bell pepper, and red bell pepper. Saute until crisp-tender, about 2 minutes. Transfer the ham and vegetables to a bowl. Wipe out the pan with a paper towel.

In the clean pan, heat 1 teaspoon butter over medium-high heat until it just begins to bubble and foam, but does not brown. Add the eggs and use a spatula to pull the cooked curds away from the edges and towards the center, allowing the raw egg to run underneath.

Tilt the pan to help move the uncooked eggs to the bottom. Create curds around the entire omelet, this will take about 45 to 60 seconds. Once the eggs are mostly cooked, similar to soft scrambled eggs with a glossy and slightly wet surface, turn the heat down to low.

Add the ham and vegetables to one half of the omelet, then sprinkle on the cheddar cheese. Turn off the heat and move the pan to a cool part of the stove. Cover the omelet and allow it to finish cooking, about 1 to 2 minutes. Using a spatula, fold the other half of the omelet on top. The surface will be lightly golden brown.

Transfer the omelet to a plate and garnish with green onions. Serve immediately.