Heat 1 tablespoon of the butter in a 10-inch skillet over medium heat. Add the mushrooms, 1/2 teaspoon salt and a few grinds of pepper and cook, stirring occasionally, until the mushrooms are dry, about 4 minutes. Transfer the mushrooms to a small bowl.
To the same skillet, add the oil. Add the shallots and cook over medium heat until they brown and soften, about 3 minutes. Add the spinach with whatever water clings to the leaves and season with salt and pepper. Increase the heat to high and stir, incorporating any browned bits from the bottom of the pan; cover and cook until the spinach is wilted, about 1 minute.
Lay a tortilla on a work surface and top evenly with 1/2 cup of the Monterey Jack. Add one-quarter of the mushrooms, one-quarter of the spinach mixture and a few dashes of hot sauce, depending on how spicy you like it. Fold the tortilla in half to enclose the fillings. Repeat with the remaining tortillas and ingredients.
Melt 1/2 tablespoon of the remaining butter in a large nonstick or cast-iron skillet over medium heat. Cook a quesadilla, flipping once, until the cheese is melted and it is golden brown on both sides, about 3 minutes per side. Transfer to a cutting board and repeat with the remaining butter and quesadillas.
Cut into triangles and serve topped with sour cream, chopped scallions, guacamole and salsa.
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