Combine beans, cheese and 1/4 cup salsa in a medium bowl. Place tortillas on a work surface. Spread 1/2 cup filling on half of each tortilla. Fold tortillas in half, pressing gently to flatten.

Heat 1 teaspoon oil in a large nonstick skillet over medium heat. Add 2 quesadillas and cook, turning once, until golden on both sides, 2 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. Serve the quesadillas with avocado and the remaining salsa.