Preheat grill to medium-high (400°F to 450°F). Generously coat peach and onion wedges with cooking spray. Grill peach and onion wedges, uncovered, until charred on both sides, 1 to 2 minutes, and about 4 minutes, respectively; set aside. When onions are slightly cool, separate into “petals.”
Place basil, water, avocado oil, vinegar, and 1 ounce of the goat cheese in a blender; process until smooth, about 1 minute. Set aside.
Roll each dough portion into a 5- to 6-inch circle on a lightly floured surface. Coat dough with cooking spray on both sides; grill over medium-high, uncovered, until golden brown and stiff, about 1 minute per side. Remove from heat. Reduce grill temperature to medium (350°F to 400°F).
Spread basil mixture evenly over grilled dough rounds, leaving a 1/2-inch border. Top evenly with grilled peaches and onion. Crumble remaining 3 ounces goat cheese evenly over tops. Grill over medium, covered, until goat cheese is warmed, about 1 minute. Remove from heat. Sprinkle with salt and pepper, and serve.