In the meantime, heat garlic cloves and oil in a small skillet over medium heat. Once garlic starts to sizzle, reduce heat to low and continue to cook, turning cloves once or twice, until soft and golden, 5 to 7 minutes. Remove from skillet and mash to a paste. Stir into ricotta along with ¼ cup of the chives; season to taste with salt and pepper
Pour remaining garlic oil, ¼ cup of water and a sprinkling of salt in a large skillet; add carrots and asparagus; cover, turn burner on high, and cook until water evaporates and vegetables are tender crisp, about 5 minutes.
Turn all burners of a gas grill on high. Without punching or kneading dough (which makes stretching more difficult), turn dough onto a lightly floured surface. Using a dough scraper or sharp knife, cut dough into 4 pieces. Working one at a time, stretch each portion into a rustic 15- by 5 to 6-inch rectangle; transfer to one of 2 large cornmeal-coated baking sheet.
Lift stretched pieces of dough onto grill. Cover and cook until bottoms are spotty brown, moving them around to ensure even cooking and piercing puffing dough as necessary, 3 to 4 minutes. Turn them over and continue to bake until lightly brown, 1 to 2 minutes longer. Remove pizza crusts from grill and reduce grill heat to low heat.