Melt 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the mushrooms, 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until browned and tender, 5 to 7 minutes. Add 1 tablespoon butter, the red onion and garlic and cook, stirring occasionally, until softened, about 3 minutes. Remove the vegetables to a plate; let cool. Wipe out the skillet.
Toss the cheese with the oregano in a large bowl. Sprinkle on one side of each tortilla and top with the mushroom mixture. Fold the tortillas in half to cover the filling.
Return the skillet to medium heat. Working in batches, cook the quesadillas, flipping once, until the cheese melts and the tortillas are browned and crisp, 5 to 7 minutes.
Meanwhile, melt the remaining 1 tablespoon butter in a medium saucepan over medium heat. Add the jalapeno and cook, stirring, until softened, about 2 minutes. Add the corn, lima beans, 1/2 cup water and a pinch of salt. Increase the heat to medium high and bring to a boil. Reduce to a simmer and cook, stirring occasionally, until most of the liquid is evaporated, about 5 minutes. Season with salt and stir in the scallions. Cut the quesadillas into wedges and serve with the succotash.