Heat the broiler and broil the flatbreads 4 to 5 inches from the heat until lightly toasted, 3 to 4 minutes per side.
Spread the sauce over the bubbly side of the flatbreads. Scatter on the spinach, ham, and cheese. Use a spoon or fork to make two hollows in the center of each flatbread for the eggs. Carefully crack each egg into a spouted measuring cup and then slowly pour it into one of the hollows.
Carefully return the pizzas to the broiler and broil them until they are crisp and the eggs are set, 6 to 8 minutes. Sprinkle on the pepper and then cut each pizza into four wedges, allowing half an egg per wedge.
Tip: If you prefer the eggs scrambled, beat them first and scramble them in a skillet before scattering them on the pizza over the tomato sauce.