Peel and thinly slice the onion. Place a large nonstick skillet over medium-high heat with 1 tbsp vegetable oil. Once hot, add the sliced onion and cook, stirring often, until browned in places, about 5 to 7 minutes. Add ½ cup water and a pinch of salt. Reduce heat to medium-low and cook until softened and lightly browned, about 3 to 5 minutes.
Halve the lime. Combine the juice from just half the lime, sour cream, cumin, and a pinch of salt in a small bowl. Stir the cumin sour cream. Peel and dice the sweet potato. Drain and rinse the black beans. Cut the remaining lime half into wedges.
Add the diced sweet potato and ¼ cup water to the skillet with the onions. Cook, stirring occasionally, until the sweet potato is tender, about 6 to 8 minutes.
Lay the corn tortillas flat on a work surface. Sprinkle just half of the mozzarella onto the tortillas. Top with the sweet potato and caramelized onions and black beans. Sprinkle with the remaining mozzarella, top each with a second tortilla, and press gently to seal.
Wipe the skillet clean and return to medium heat with 1 tsp vegetable oil. Once hot, add the first quesadilla. Cook until the bottom is browned, 2 to 3 minutes, and carefully flip to brown the other side, about 1 to 2 minutes more. Transfer to a plate. Add another 1 tsp vegetable oil to the skillet, and repeat with the second quesadilla.
Cut quesadillas into quarters. Serve the sweet potato & black bean quesadillas with cumin sour cream and lime wedges. Serve any remaining filling on the side.