The Ultimate Summer Pizza

Name: The Ultimate Summer Pizza
Ingredients: 1/2 recipe Pizza Dough for Grilling (split into 2 pieces for rising), or 1 to 1 1/4 pounds store-bought pizza dough; 8 ounces low-moisture whole or skim-milk mozzarella cheese, shredded (about 2 cups); 4 ounces feta cheese; 1 handful fresh basil sprigs; 1 large clove garlic; 1/3 cup olive oil, plus more for the grill grates; 3/4 teaspoon kosher salt, divided; 1/4 teaspoon red pepper flakes (optional); 1/4 medium red onion; 1 cup grape or cherry tomatoes (5 to 6 ounces); 1 ear fresh corn (about 12 ounces), or 3/4 cup frozen corn kernels; 1 small yellow summer squash or zucchini (5 to 6 ounces); All-purpose flour, for dusting