Wash and chop your chosen vegetables into small, bite-sized pieces.
In a small bowl, crack the two eggs. Add the milk, salt, and pepper. Whisk everything together really well until combined and slightly frothy.
Place a non-stick skillet over medium heat. Add the olive oil or butter and let it melt and heat up. Add your chopped vegetables to the skillet. Cook for about 3-5 minutes, stirring occasionally, until softened and fragrant.
Reduce the heat to low-medium. Pour the whisked egg mixture evenly over the vegetables in the skillet. Let the eggs cook undisturbed for about 1-2 minutes, until the edges start to set.
Sprinkle the shredded cheese over one half of the omelet. Continue to cook for another 1-2 minutes, until the omelet is mostly set but still slightly soft in the center.
Using a spatula, gently fold the cheesy half of the omelet over the plain half. Cook for another minute to make sure the cheese is melted and the omelet is heated through. Slide the Veggie Omelet out of the skillet onto a plate.
Toast your bread slices to your desired level of crispness. Once toasted, you can add butter or your favorite spread if you like.
Place the warm Veggie Omelet next to your toast on a plate. Serve immediately.