In a bowl combine black beans, red onions, tomatoes, cilantro, chiles and lime juice. Season to taste with salt and pepper.
Combine chèvre and grated cheese together in a small bowl.
Preheat a 10-inch or larger skillet on medium heat. Spread softened butter or oil on one side of two tortillas. Place one tortilla, butter-side down, in the hot skillet. Spread 1/4 of the cheese mixture on top of tortilla and top with 1/4 of the black bean mixture. Place the other tortilla, butter side up, on top.
Cook until the bottom tortilla is golden brown; using a pancake turner, carefully flip the quesadilla over and cook until golden brown on the other side. Repeat process 3 more times with remaining tortillas and filling.
To serve, cut each quesadilla into 8 wedges and serve with a dollop of sour cream and salsa on the side.