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.