Preheat oven to 400 degrees

Bring a medium pot of lightly salted water to a boil

Prepare a baking sheet with foil and cooking spray

Stem and mince parsley.

Cut potato into ¾" cubes. Peel, trim, and cut carrot into ½" pieces. Pat chicken breasts dry, and season both sides with ¼ tsp. salt and ¼ tsp. pepper.

Place a large oven-safe non-stick pan over medium-high heat and add 2 tsp. olive oil. Add panko to hot pan and toast until light golden brown, 30-60 seconds.

Remove from burner. Let panko cool 5 minutes. Wipe pan clean and reserve. Combine cooled panko, goat cheese, parsley (reserve a pinch for sauce), ¼ tsp. salt, and ¼ tsp. pepper in a mixing bowl. Set aside.

Return pan used to toast panko to medium-high heat and add 2 Tbsp. olive oil. Add potatoes to hot pan and stir, 1 minute. Cover, reduce heat to medium, and cook 5 minutes, stirring halfway through. Uncover, place pan in hot oven, and roast until golden brown and tender, 20-22 minutes, stirring halfway through. Transfer cooked potatoes to a plate and season with a pinch of salt and pepper. Wipe pan clean and reserve. While potatoes roast, start chicken.

Place chicken breasts on prepared baking sheet. Top each chicken breast evenly with goat cheese-panko mixture, pressing gently to adhere. Roast until well-browned and chicken reaches a minimum internal temperature of 165 degrees, 17-20 minutes. Remove from oven and rest 5 minutes. While chicken roasts, make carrot mash.

Add carrot pieces to boiling water and cook until very tender, 15-18 minutes. Drain carrot in a colander. Return to pot and add 1 Tbsp. evaporated milk (reserve remaining for sauce) and 1 tsp. olive oil. Mash until carrot forms a coarse puree. Season with a pinch of salt and pepper. Set aside and keep warm.

Return pan used to cook potatoes to medium heat and add 1 tsp. olive oil. Add garlic to hot pan and cook until fragrant, 30 seconds. Add remaining evaporated milk and bring to simmer. Stir often until slightly thickened, 3-4 minutes. Remove from burner and swirl in butter and remaining parsley.