1. Preheat a charcoal or gas grill for direct, medium-high grilling. If using charcoal, push 2/3 of hot coals to 1 side of grill; spread remaining 1/3 on other side. If using a gas grill, reduce heat on 1 side to medium.
2. Dust a 14-inch-wide pizza peel with flour or cornmeal; generously flour a work surface. Roll dough into a 12-inch round. Transfer dough to pizza peel; brush with 1 Tbsp. oil.
3. Put mozzarella within reach of the grill. Slide dough onto hottest part of grill; cover grill. Cook, rotating as needed, until bottom is blistered and browned, about 2 minutes. Flip dough onto cooler side of grill, cooked side facing up.
4. Arrange mozzarella on crust. Cover grill; cook until bottom has browned and cheese has melted, 2 to 3 minutes, occasionally checking bottom of crust to ensure it's not burning. If it is, reduce heat (if using a gas grill) or move off the grill (if using charcoal). Transfer pizza to a cutting board.
5. In a bowl, combine arugula and tomatoes. Drizzle with remaining 2 tsp. olive oil and balsamic vinegar and toss to coat; season with salt and pepper. Pile salad on top of pizza, then slice and serve.