Heat oven to 425°. Brush a 10- by 15-inch rimmed baking sheet with 1 tsp. olive oil. Press dough into pan; pre-bake crust 5 minutes.
In a large bowl, beat eggs, milk, and 1 Tbs. pesto; season with salt and pepper. Heat remaining 2 tsp. oil in a large nonstick skillet over medium heat. Add egg mixture; softly scramble, removing pan from heat when eggs are still a bit undercooked.
Spread remaining 4 Tbs. pesto over crust, leaving a 1/2-inch border. Top with eggs, then mozzarella and tomato slices. Bake until cheese is melted and crust is golden, about 10 minutes. Scatter basil over top.