In a small bowl, mix 1/2 cup EVOO, the garlic and crushed red pepper. Season well with salt and set aside.
Preheat a grill to 500 degrees . Place a heatproof skillet on the grill; add the remaining 1 tbsp. EVOO and the tomatoes. Cook the tomatoes until charred in spots and softened, shaking the pan occasionally, about 5 minutes. Transfer to a medium bowl; mix in 1 tbsp. of the red pepper oil. Season with salt.
Lower the heat on one side of the grill to low. Oil the grate on that side of the grill. Move the shaped dough onto the oiled grill. Lower the edge of the dough onto the oiled grill, draping the dough perpendicular to the grate. For 9-inch pizzas, grill 2 crusts at a time. For 3- to 4-inch pizzas, grill 8 crusts at a time.
Cover the grill and cook until the crusts begin to bubble and brown, about 3 minutes. Loosen the crusts with a metal spatula; cover and cook for 1 minute more. Flip the crusts over; cook until beginning to brown on the bottom, about 2 minutes.
Working in batches, invert the crusts flat side down onto a baking sheet. Brush with some of the red pepper oil, then divide the mozzarella, goat cheese, tomatoes and olives evenly among them. Place the baking sheet on the cooler side of the grill; cook until the mozzarella melts, 2 to 3 minutes. Sprinkle with the sliced basil. Drizzle with more red pepper oil and garnish with basil leaves.