Core the apples and slice into rounds about 1/4 inch thick (use a mandolin or food processor if you have one).
Slice the eggplant into rounds about 1/4 inch thick.
Heat the olive oil and butter in a large skillet over medium-high heat. Add the eggplant and apple and cook about 5 minutes until softened (you can do this in 2 batches if your skillet isn't large enough to cook it all at once). Set aside.
Butter one side of a tortilla and place butter side down on a baking sheet. Repeat with 3 additional tortillas.
Distribute the apples, eggplant, spinach and cheese among the 4 tortillas. Sprinkle each with 1/4 teaspoon nutmeg.
Butter each of the remaining 4 tortillas on one side, and place on top of the quesadillas, butter side up.
Bake for about 10 minutes, until the cheese is melted.