Preheat the oven to 400°F. Line a baking sheet with parchment paper.

Cut each squash half crosswise into ¼ inch slices, resembling halfmoons. In a large bowl, toss the squash with the olive oil, and sprinkle with salt and pepper to taste. Place the squash on the prepared baking sheet and roast for 20 to 25 minutes, turning the pieces over halfway through, or until the squash feels soft when pierced with a fork and the slices are slightly caramelized. Set aside to cool slightly.

In a small bowl, mash the butter and Parmesan together. To assemble: Spread the Parmesan butter on one side of each tortilla and place them on your work surface, buttered side down. Place 6 slices of squash on the bottom half of each tortilla, followed by the Gruyére, blue cheese, and sage. Fold the tortillas in half to enclose the filling, forming a half-moon. (You may have extra squash slices; save for another use or enjoy as a snack.)

Stovetop method: Heat a large nonstick skillet over medium heat for 2 minutes. Place 2 of the quesadillas in the skillet, cover, and cook for 2 minutes or until the undersides are golden brown and the cheese has begun to melt. Watch carefully, as the Parmesan butter can burn easily.

Uncover, and turn the quesadillas with a spatula. Cook for about 1 minute, or until the undersides are golden brown and the cheese has melted completely. Repeat with the remaining 2 quesadillas. Cut the quesadillas in half, and serve immediately.

Sandwich maker method: Preheat the sandwich maker (if your machine has adjustable heat, set it to medium). Follow directions for quesadilla assembly. Cook for 3 to 4 minutes or until the cheese has melted completely. Watch carefully, as the Parmesan butter can burn easily. Unless you have a large sandwich maker, you will have to cook the quesadillas in batches.

Gas grill method: Brush the grill rack with oil and preheat the grill to medium. Follow directions for quesadilla assembly. Put the quesadillas on the grill and follow directions for the stovetop method.