Place puﬀ pastry on counter 40 minutes to thaw. Preheat oven to 350°. Meanwhile, in medium saucepot, melt butter over medium heat. Add onion, salt and pepper, and cook 3 minutes, stirring occasionally. Reduce heat to medium-low and cook 15 to 18 minutes longer or until golden brown and tender, stirring occasionally. Stir in cherries, vinegar, barbeque sauce and thyme, and cook 2 minutes. Fold in chicken and remove from heat.
Spray 9 standard muﬃn cups with nonstick cooking spray. Cut puﬀ pastry sheet into 9 equal-size squares. Fit pastry squares into prepared muﬃn cups with corners of pastry squares sticking up; gently press squares onto bottom and sides of cups.
Evenly divide onion mixture into each pastry square-lined cup. Bake tartlets 15 minutes. Evenly divide cheese over top of each tartlet and bake 3 to 5 minutes longer or until cheese melts and tartlets are golden brown.