Preheat oven to 500 degrees F.
Fill a large shallow saucepan with about 3 inches of water and 1 tablespoon salt, bring to a gentle simmer. Brush 2 baking sheets with 3 tablespoons of the olive oil. With a sharp knife, chop tomatoes, sprinkle with 1/4 teaspoon salt, place in a sieve, and drain 5 minutes; discard juice.
In a small bowl, mix tomatoes with 1 tablespoon oil. Cut pizza dough in 4 pieces. On oiled baking sheets, pat and stretch each one to make an 8-inch round. On each pizza round, spread tomatoes and sprinkle with mozzarella. Bake until dough is browned and cheese is melted, about 12 minutes.
Break an egg into a small bowl; place the bowl close to the simmering water and slide the egg into the water. With a large spoon, nudge the egg white toward the yolk; repeat with the other eggs. Turn off the heat, cover and cook just until the white sets and the yolk is still soft, about 2 minutes.
Meanwhile, place each pizza on a dinner plate. Arrange Prosciutto di Parma strips on top; sprinkle with basil ribbons. Lift each egg with a slotted spoon and set on top of a pizza; drizzle with remaining olive oil and sprinkle with black pepper if desired.