Preheat grill to medium-high heat (350° to 400°). Heat 1 tablespoon oil in a large skillet over medium-high heat. Add mushrooms; sauté 3 minutes. Add onion; sauté 4 minutes. Add chicken and chicken stock; cook 1 minute.
In a small bowl, combine ricotta cheese, salt, and pepper.
Sprinkle semolina flour evenly over a large, rimless baking sheet. Place pizza dough on baking sheet, and pat into a 14-inch circle.
Brush grill grates with oil, and slide pizza dough onto grates. Brush top of dough with remaining 1 tablespoon oil. Grill 2 minutes or until grill marks appear. Reduce heat to medium-low; flip dough, and spread evenly with pesto, leaving a 3/4-inch border. Spread chicken mixture evenly over pesto. Top evenly with mozzarella and tomato slices, and dot with ricotta mixture. Grill, covered with grill lid, 5 to 7 minutes or until cheese melts. Remove pizza from grill, and top with basil. Serve immediately.