1Combine barbecue sauce, tomato paste, vinegar and chipotle in a medium bowl.
2Heat 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.
3Place tortillas on a work surface. Spread 3 tablespoons cheese on half of each tortilla and top with one-fourth (about ½ cup) of the filling. Fold tortillas in half, pressing gently to flatten.
4Heat 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.
Ingredient note: Chipotle chiles in adobo sauce are smoked jalapeños packed in a flavorful sauce. Look for the small cans with the Mexican foods in large supermarkets. Once opened, they'll keep up to 2 weeks in the refrigerator or 6 months in the freezer. Ground chipotle, made from dried smoked jalapeños, can be found in the specialty-spice section of most supermarkets or online at penzeys.com.