Combine barbecue sauce, tomato paste, vinegar and chipotle in a medium bowl.
Heat 1 tablespoon oil in a large nonstick skillet over medium heat. Add mushrooms and cook, stirring occasionally, for 5 minutes. Add onion and cook, stirring, until the onion and mushrooms are beginning to brown, 5 to 7 minutes. Transfer the vegetables to the bowl with the barbecue sauce; stir to combine. Wipe out the pan.
Place tortillas on a work surface. Spread 3 tablespoons cheese on half of each tortilla and top with one-fourth (about 1/2 cup) of the filling. Fold tortillas in half, pressing gently to flatten.
Heat 1 teaspoon oil in the pan over medium heat. Add 2 quesadillas and cook, turning once, until golden on both sides, 3 to 4 minutes total. Transfer to a cutting board and tent with foil to keep warm. Repeat with the remaining 1 teaspoon oil and quesadillas. Cut each quesadilla into wedges and serve.