Cut green onions crosswise into diagonal slices. Finely chop oregano. Heat olive oil in 9-inch nonstick skillet, over medium heat. Add green onions and cook until slightly softened, about 1 minute. Add spinach and cook, stirring, until leaves wilt, about 2 minutes. Drain any liquid from pan, then stir in oregano.

In a small bowl, whisk together eggs and milk. Sprinkle feta onto spinach mixture, then pour egg mixture on top. Using a spatula, softly stir mixture in pan until eggs are set, about 4 minutes.

Place mozzarella and pepper slices down the center of the egg mixture. Gently fold over one side to cover filling, then fold in other side. Slide omelet onto a flat plate and serve promptly.