Peel apple, quarter, and remove core. Cut into ½" dice. Mince garlic. Stem oregano and coarsely chop. Pat chicken breasts dry, and season both sides with a pinch of salt and pepper.
Start the Vegetables
Place a medium non-stick pan over medium-high heat and add 2 tsp. olive oil. Add butternut squash, apple, and half the garlic (reserve remaining for sauce) to hot pan and stir occasionally until lightly browned, 4-6 minutes. Remove from burner and toss with half the oregano, half the butter (reserve remaining of both for sauce), ¼ tsp. salt and a pinch of pepper. Transfer vegetables to prepared casserole dish (you may also use two small ramekins). Wipe pan clean and reserve.
Sear the Chicken
Return pan used to cook vegetables to medium-high heat and add 1 tsp. olive oil. Add chicken breasts to hot pan and sear until golden brown, 4-5 minutes per side. Transfer chicken to a plate. Chicken will finish cooking in a later step.Reserve pan; no need to wipe clean.
Finish the Gratin
Top vegetables in casserole dish with goat cheese and roast until squash tender and cheese is lightly melted, 8-10 minutes. While gratin cooks, make sauce.
Make Sauce and Finish Dish
Return pan used to cook chicken to medium-high heat. Add ½ tsp. olive oil and remaining garlic and cook until aromatic, 30-60 seconds. Add ½ cup water, brown sugar, remaining oregano, chipotle powder (to taste- it's quite spicy) and chicken and any accumulated juices to pan. Flip chicken occasionally until sauce is thick enough to coat the back of a spoon and chicken reaches a minimum internal temperature of 165 degrees, 3-5 minutes. Remove from burner and swirl in butter. Place chicken on a plate and place gratin next to chicken. Top chicken with sauce.