Preparation
Preheat oven to 450°F.
In a large nonstick skillet, heat 1 teaspoon of oil over medium heat. Add spinach, in batches if necessary, and cook until wilted, 2 to 3 minutes.
Brush inside of each pita round with 1 teaspoon oil. Place pita rounds, oiled side up, on a large baking sheet and bake until starting to brown, about 5 minutes. Remove from the oven.
Divide tomatoes and spinach evenly among pita halves, leaving an empty space in the center of each for an egg. Crack 1 egg into the center of each pita. Sprinkle with salt and pepper, return to the oven, and bake until yolks are lightly set, 8 to 10 minutes. Sprinkle with cheese and continue baking until cheese has softened, about 2 minutes more. Serve warm.