In 10-inch skillet, heat oil over medium heat. Add onion; cook 10 minutes, stirring occasionally. Reduce heat to medium-low; cook 5 to 10 minutes longer, stirring frequently, until onions are very tender and golden. Remove from heat; set aside.
Heat gas or charcoal grill for indirect cooking as directed by manufacturer. Spray 2 large cookie sheets with cooking spray. Unroll both cans of dough; cut each rectangle of dough into 4 equal rectangles. Spray both sides of each rectangle with cooking spray; place on cookie sheets.
Place 2 or 3 dough rectangles at a time directly on grill rack on unheated side of two-burner gas grill or over drip pan on charcoal grill. (If using one-burner gas grill, cook over low heat.) Cover grill; cook 5 to 7 minutes or until edges of dough look dry (check occasionally to make sure bottoms of crusts are cooking evenly; rotate crusts if necessary). Cook about 2 minutes longer or until bottoms of crusts are golden and have grill marks. Using large pancake turner, remove crusts from grill to cookie sheet. Repeat with remaining dough rectangles.
Turn cooked crusts over so cooked sides are up; spread evenly with pizza sauce. Top each with sausage, onions and cheese.
Heat oven to 200°F. Return 4 pizzas at a time to grill rack over indirect heat. Cover grill; cook about 5 minutes or until bottoms are golden brown and cheese is melted. Remove cooked pizzas from grill to cookie sheet; place in oven to keep warm while cooking remaining pizzas. Sprinkle basil over pizzas.