Heat olive oil in a large skillet over medium heat; cook and stir asparagus and red bell pepper until the vegetables start to soften, about 3 minutes. Stir in cherry tomatoes, spinach, garlic, oregano, basil, and salt and continue cooking until tomatoes are soft and spinach has cooked down, another 3 to 5 minutes. Remove from heat and transfer vegetables to a plate.
Melt butter in clean skillet over medium heat. Whisk eggs and milk in a bowl and pour into hot butter, swirling skillet to cover entire bottom with egg mixture. Pull up an edge of the omelet with a spatula and tilt pan to allow unset egg to run underneath and cook; continue around pan, lifting omelet edge and tilting pan, until all the egg mixture is set. Sprinkle omelet with salt.
Spoon cooked asparagus mixture onto one side of the omelet and sprinkle with feta and Cheddar cheeses. Gently fold half the omelet over the vegetables and cheese and press edges lightly to seal in the filling. Cook until filling is hot and Cheddar cheese has melted, 1 to 2 more minutes. Cut in slices to serve.