Melt 2 tablespoons butter in large skillet over medium-high heat. Add onion and sauté for 5 minutes. Add sliced mushrooms, sprinkle with salt and pepper, and sauté for 2 minutes. Add wine and continue cooking until liquid is evaporated and the onions are translucent, 6–8 minutes. Reduce heat to medium-low and add spinach, stirring gently until wilted. Remove mixture from heat.
Heat waffle maker at a medium-high setting. Lightly butter one side of each tortilla. To assemble quesadillas, place a tortilla, butter-side down, on the bottom of the waffle maker. Layer the tortilla with grated cheese, the spinach-mushroom mixture, and then a little more cheese. Top with the second tortilla, butter-side up. Close the waffle maker and cook until the cheese melts and the tortilla is golden brown, about 4 minutes.
Repeat with remaining tortillas and filling. Slice quesadillas into four wedges and serve with salsa and sour cream.