Preheat oven to 400; line a baking sheet with parchment paper.
Cut each squash half crosswise into 1/4-inch slices.
In a big bowl, toss the squash with the olive oil; sprinkle with salt and pepper.
Place the squash on the prepared baking sheet; roast for 20-25 minutes, turning 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.
Spread the parmesan butter on 1 side of each tortilla and place them on a work surface, buttered side down.
Place 6 slices of squash on the bottom half of each tortilla, followed by the Gruyere, blue cheese, and sage; fold the tortillas in half to enclose the filling.
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 in half and serve immediately.