01 -
Add the eggs, ⅛ teaspoon salt, and water to a medium bowl. Beat with a wire whisk or fork until whites and yolks are completely combined with no spots of egg white remaining. The mixture should flow evenly off the whisk. Allow to rest at room temperature for 10-15 minutes.
02 -
Melt 1 tablespoon of butter in a 9- or 10-inch nonstick skillet over medium heat. Add the onion, green and red peppers, season with salt, and cook until softened, about 5 minutes. Add the ham and cook until browned, 2-3 minutes. Transfer the vegetable mixture to a bowl.
03 -
Wipe the skillet clean, then add the remaining tablespoon of butter and melt over medium-high heat. Cook until the butter starts to brown.
04 -
Pour in the egg mixture. As the eggs start to set, use a rubber spatula to push from the edges toward the center of the pan. Tilt the pan as needed to let uncooked eggs flow under the cooked portion. Continue until eggs are mostly set, about 1 minute. The top should still look wet, but not runny.
05 -
Remove the pan from heat and sprinkle the vegetable mixture and cheese over half of the omelet. Cover the pan with a lid and let sit for 1-2 minutes, until the cheese is melted and the omelet looks fluffy and cooked through.
06 -
Run a rubber spatula around the edge of the pan to loosen the omelet. Gently fold in half, then slide onto a serving plate. Serve immediately.