Heat oil in a large nonstick skillet over medium heat. Add onion, sugar, and salt; sauté 3 minutes. Reduce heat to medium- low, and cook 20 minutes or until very golden brown, stirring frequently. Stir in 1/2 teaspoon thyme.
Combine ricotta cheese, Asiago cheese, garlic, and pepper; stir well. Set aside.
Coat pitas with cooking spray. Place on grill rack; cover and grill 2 minutes on each side or just until crisp. Remove pitas from grill. Spread cheese mixture evenly over warm pitas; top evenly with caramelized onion.
Cover grill rack with foil, and place pitas on foil. Cover and grill 2 minutes or until pitas are thoroughly heated. Cut each pita into 4 wedges, and serve immediately.