Heat oven to 425 degrees. On a parchment-lined baking sheet, brush flatbread with 1 tablespoon oil. Toss spinach and onion with remaining oil and lemon juice; season with salt and pepper. Mound mixture on flatbread and bake 5 minutes.
Crack eggs over spinach, season with salt and pepper, and bake until eggs are just set, 6 to 7 minutes more.