Preheat the oven to 375 degrees F.
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 the juice from half a lime to the pan. Cook until firm and golden brown, about 6 minutes per side; after the first 6 minutes, squeeze the remaining lime half over the chicken and turn.
In a large nonstick skillet over medium heat, cook jalapeno slices for 4 minutes on each side. Remove, and set aside. Prepare the 2 quesadillas: Crumble 4 ounces goat cheese on 2 flour tortillas. Place about 10 cilantro leaves, 8 pieces of the roasted squash, and 4 slices of jalapeno on each tortilla. Cover with another tortilla.
Reheat the skillet over medium-high heat with 1 teaspoon olive oil. Cook each quesadilla until golden in color, about 1 1/2 minutes per side. Slide quesadilla onto a dinner plate; turn plate over, and return quesadilla to the pan, uncooked side down. Remove from heat, and cut each quesadilla into quarters with kitchen scissors or a knife.
Serve 1 breast, cut in half, with 2 quesadilla quarters. Garnish with any leftover squash pieces and a few sprigs of cilantro.