Butternut is a particularly versatile squash, adapting itself to other ingredients without ever losing its identity. In fact, it's almost as adaptable as chicken, so it makes sense that the two should work so well together.
Heat oven to 375 degrees. Drizzle 1 1/2 tablespoons olive oil in the bottom of a 9-by-13-inch baking dish. Place the squash pieces on top. Season with salt and pepper to taste and a pinch of red-pepper flakes. Roast in the oven until fork-tender, about 40 minutes. Remove from oven, and let cool.
Meanwhile, prepare chicken: Season each breast with salt and pepper. Place 1/4 teaspoon garlic and 2 slices of lime under the tender of each breast. (The tender is the natural pocket on the underside of the breast.) In a large cast-iron skillet over medium heat, melt butter and 1 teaspoon olive oil. Place chicken breasts in the skillet, tender side up. Add the remaining 1 tablespoon minced garlic, cumin, and half the lime juice to the pan. Cook until firm and golden brown, about 6 minutes per side; after the first 6 minutes, pour the remaining lime juice over the chicken, and turn chicken.
In a large nonstick skillet over medium heat, cook jalapeno slices for 4 minutes on each side. Remove, and set aside. Prepare the two quesadillas: Crumble 4 ounces goat cheese on 2 flour tortillas. Place about 10 cilantro leaves, 8 pieces roasted squash, and 4 jalapeno slices on each tortilla. Cover with another tortilla.
Reheat the skillet with 1 teaspoon olive oil over medium-high heat. Cook quesadilla until golden in color on the bottom, about 1 1/2 minutes. Slide quesadilla onto a dinner plate; turn over, and return quesadilla to the pan, uncooked side down. Cook until golden on the bottom. Remove from heat, and cut quesadilla into quarters with kitchen scissors or a knife. Repeat with the remaining quesadillas, adding more oil as necessary.
Serve one breast, cut in half, with 2 quesadilla quarters. Garnish with any leftover squash pieces and a few sprigs cilantro.