Place the black beans, jalapeño, garlic, salt, and cilantro in a food processor. While running, slowly add the chicken broth to make a smooth puree.
Brush the olive oil on the bottom side of one tortilla. Spread 1/8 of the black bean puree on the tortilla. Sprinkle with 3/4 cup cheese, then spread with an additional 1/8 of the black bean puree.
Spoon 1/4 cup of salsa on top of the cheese and cover with the top tortilla. Brush with olive oil. Repeat with remaining ingredients.
Preheat a grill to medium heat or 350 degrees F (175 degrees C).
Place the quesadillas on the grill two at a time, unless your grill is very large. Grill for 3 to 4 minutes, then give a quarter turn. Cook for 2 additional minutes, then flip the tortilla over.
Cook for 2 minutes on the other side, then give a quarter turn. Check to see if the cheese has melted.
Let stand for 3 to 4 minutes before cutting into 8 wedges.