Preheat grill to medium-high (450°). Divide pizza dough into 2 balls and let rest at room temperature until no longer stiff, 20 to 30 minutes.
Meanwhile, blend garlic, pepper, 2 tbsp. oil, and 1/2 tsp. salt in a medium bowl.
Heat remaining 2 tbsp. oil in a large frying pan over medium heat. Add leeks and remaining 1/4 tsp. salt and cook, stirring often, until softened, about 8 minutes. Add green onions and cook, stirring often, until wilted, about 3 minutes. Stir into bowl of garlic oil.
Stretch each ball of dough into a 12-in. round and place on a lightly oiled sheet of parchment paper. Flip each dough round onto preheated grill, then remove parchment paper. Grill, covered, until dough puffs, about 2 minutes. Transfer rounds to a baking sheet, grilled sides up.
Sprinkle grilled side of each round with 1 cup cheese, then half of onion mixture and half of remaining cheese. Using a wide spatula, transfer pizzas to grill and cook, covered, 1 to 2 minutes; then pop any large bubbles (this keeps the pizza fairly flat so toppings stay on). Cook until browned and crisp on bottoms, 3 to 5 minutes more. Remove pizzas from grill and sprinkle with chives.