Preheat your oven to 400°F (200°C). Pierce the sweet potatoes several times with a fork, rub them with the olive oil, and place them on a baking sheet. Roast for 45 to 50 minutes, or until they are very tender when pierced with a knife. Let them cool until they are safe to handle, then scoop the flesh into a medium bowl and discard the skins.
Mash the sweet potato flesh with a fork until mostly smooth. Stir in the kosher salt, black pepper, drained black beans, Monterey Jack cheese, cheddar cheese, red onion, cilantro, cumin, and chili powder until all ingredients are well combined.
Lay the four tortillas on a clean work surface. Divide the sweet potato and bean mixture equally between them, spreading it evenly over one half of each tortilla and leaving a 1-inch border around the edge. Fold the bare half of each tortilla over the filling to create a half-moon shape.
Heat a large skillet or griddle over medium heat. Lightly spread about 1/2 tablespoon of the softened butter onto one side of the skillet. Place one or two quesadillas in the skillet, buttered-side down, and cook for 3 to 4 minutes, until the bottom is golden brown and crisp.
Carefully flip the quesadillas with a spatula. Cook for another 3 to 4 minutes on the second side, until it is also golden brown and the cheese inside is fully melted. Repeat with the remaining quesadillas, adding more butter to the skillet for each batch.
Transfer the cooked quesadillas to a cutting board. Let them rest for 2 minutes before slicing each one into three wedges. This allows the filling to set slightly for cleaner cutting.