Pat chicken breasts dry, and season both sides with ¼ teaspoons salt and a pinch of pepper. Place a medium non-stick oven-safe pan over medium heat. Add butter to pan and let melt. Add chicken breasts, skin side down, and cook undisturbed until skin is browned, 6-8 minutes. Flip chicken and remove from burner. While chicken sears, make mashed potatoes.
Make potatoes: Peel and cut potato into 1" dice. Bring a medium pot with potato and enough lightly salted water to cover to a boil. Reduce to a simmer and cook until potato is fork-tender, 15-18 minutes. Drain in a colander and return to pot. Add ¾ the milk (reserve remaining for sauce), Parmesan, 1 teaspoons pepper, and ¼ teaspoons salt. Mash until smooth. Cover and set aside. While potato cooks, prepare ingredients.
Prepare the Sprouts: Trim bottoms off Brussels sprouts and halve vertically (quarter if larger than a ping-pong ball). Stem and coarsely chop thyme.
Place Brussels sprouts on prepared baking sheet and toss with 1 teaspoons olive oil, ¼ teaspoons salt, and a pinch of pepper. Spread into a single layer and roast until browned and tender, 10-12 minutes. Sprinkle cooked Brussels with half the thyme (reserve remaining for sauce).
Place pan with chicken in oven and roast until golden brown and chicken reaches a minimum internal temperature of 165 degrees, 12-15 minutes. Carefully, remove chicken from pan and rest 5 minutes. Reserve pan and any pan drippings.
Make the Sauce: Very carefully (handle is hot!), return pan used to cook chicken to medium-high heat and add white cooking wine. Bring to a boil. Add remaining milk, return to a boil, and stir constantly until sauce thickens and a line drawn in sauce holds for 10-15 seconds, 3-5 minutes. Add remaining thyme and remove from burner.
Plate mashed potatoes first, and sprouts next to taters. Now add chicken. Add sauce to potatoes and chicken as desired. Bon appétit!