Place flatbreads on a baking sheet lightly coated with cooking spray; lightly coat tops of bread with cooking spray. Broil 1 minute. Remove pan from oven; turn breads over. Lightly coat with cooking spray. Broil 1 minute.
Heat a large skillet over medium heat. Add 1 teaspoon oil to pan. Add onion, salt, and red pepper; cook 3 minutes. Add spinach; cook 1 minute. Remove mixture from pan; keep warm.
Wipe pan clean with a paper towel. Heat pan over medium heat. Add remaining 2 teaspoons oil to pan; swirl to coat. Crack eggs into pan; cook 1 minute on each side or until desired degree of doneness.
Divide spinach mixture evenly among flatbreads. Top evenly with cheese and chopped prosciutto. Broil 1 minute or until cheese melts. Top each flatbread with 1 egg; sprinkle black pepper over eggs. Serve immediately.