Heat the oil over medium-high heat in a small saucepan and add the garlic. Cook for a minute or so until fragrant, then add the beans and the cumin, stirring to combine. Add the lime juice and let simmer for a few minutes to thicken, stirring occasionally. When the mixture resembles a thick paste, stir in the cilantro and season with salt and pepper. Set aside.
Arrange two tortillas on a work surface. Spread each with 1/4 cup of the black bean puree, and layer with chicken, avocado slices, and cilantro leaves. Squeeze each with a lime wedge and season with salt and pepper. Cover with the remaining two tortillas.
Heat a large non-stick skillet over medium-high heat. Brush the tops of the quesadillas lightly with olive oil. Using a large spatula, carefully place one quesadilla oiled-side down into the skillet. Let cook for 2 minutes or so until crisp and golden brown. Brush the top of the quesadilla with olive oil and using the spatula, carefully flip it. Let the remaining side cook for another 2 minutes, or until golden brown, pressing down to make sure the filling is warmed through.
Slide onto a cutting board and cook the remaining quesadilla, brushing each side with olive oil as before. Cut the quesadillas into wedges and serve with a light salad (baby arugula, avocados, thinly sliced red onion, orange slices, etc) and a blood orange cilantro mocktail.