Peel, trim, and cut carrots into ½" rounds. Cut potatoes into ¾" cubes. Mince garlic. Stem and mince parsley. Combine panko, goat cheese crumbles (breaking up with your hands if needed), parsley (reserve a pinch for sauce), 2 tsp. olive oil, 1 tsp. salt, and ½ tsp. pepper in a small bowl. Rinse chicken, pat dry, and season both sides with ½ tsp. salt and ¼ tsp. pepper.

Heat 1 Tbsp. olive oil in a large oven-safe non-stick pan over medium-high heat. Add potatoes to hot pan and cook, stirring occasionally, 1 minute. Cover, reduce heat to medium, and cook 5 minutes, stirring once halfway through. Uncover, place pan in oven, and roast 20-22 minutes, stirring halfway through, until golden brown and tender. Remove from oven, season with ¼ tsp. salt and a pinch of pepper, and transfer to a plate. Wipe pan clean and reserve for making sauce. While potatoes are roasting, start chicken.

Place chicken breasts on prepared baking sheet. Top each breast with goat cheese-panko mixture. Roast until well-browned and chicken reaches a minimum internal temperature of 165 degrees, 15-18 minutes. Remove from oven and rest 5 minutes. While chicken is roasting, make carrot mash.

Add carrots to boiling water and cook until very tender, 12-15 minutes. Drain in colander and return to pot with half the butter. Mash with a potato masher or fork to form a coarse puree. Season with ½ tsp. salt and ¼ tsp. pepper. Set aside and keep warm.

Heat 1 tsp. olive oil in pan used to cook potatoes over medium heat. Add garlic to hot pan and cook until fragrant, 30 seconds. Add evaporated milk, bring to simmer, and cook until thick enough to coat the back of a spoon, 3-4 minutes. Remove pan from burner and swirl in remaining butter and reserved parsley. Season with a pinch of salt and pepper.

Serve carrot mash and crispy potatoes on a plate. Place crusted chicken in front of vegetables and potatoes, and spoon sauce in front of chicken.