Peel and cut potato into ¾" dice. Bring a medium pot with potato and enough lightly salted water to cover to a boil. Reduce to a simmer and cook until potatoes are fork-tender, 15-18 minutes. Reserve ¼ cup potato water. Drain in a colander and transfer to a mixing bowl. Mash potatoes. Reserve pot; no need to wipe clean. While potatoes cook, prepare ingredients.
Prepare Ingredients and Roast Garlic
Set aside 3 cloves garlic. Mince 1½ cloves and thinly slice remaining ½ clove. Trim ends off green beans. Coarsely chop sun-dried tomatoes. Halve and juice lemon. Pat chicken breasts dry, and season both sides with paprika, ½ tsp. salt and a pinch of pepper. Place whole garlic cloves on a piece of foil and add ¼ tsp. olive oil and a pinch of salt. Fold foil around garlic, making a pouch. Place pouch, opening side up, directly on oven rack. Roast until soft and lightly caramelized, 15-18 minutes. While garlic roasts, sear chicken.
Cook the Chicken
Heat a medium non-stick pan over low. Add 1 Tbsp. olive oil and chicken breasts to hot pan. Cook undisturbed until browned, 3-4 minutes. Flip chicken and add ⅓ the butter. Cook until chicken reaches a minimum internal temperature of 165 degrees, 3-4 minutes. Remove chicken to a plate. Pour pan drippings into bowl containing potatoes. Reserve pan; no need to wipe clean.
Finish Potatoes and Roast Green Beans
Smash roasted garlic into a chunky paste. Heat pot used to cook potatoes to medium-high and add garlic paste, reserved water, heavy cream, and ⅓ the butter. Once liquid is boiling, whisk in potatoes and whisk until thickened, 30-60 seconds. Season with a pinch of salt and pepper and remove from burner. Add green beans and sliced garlic to prepared baking sheet. Toss with 1 tsp. olive oil and a pinch of salt and pepper. Spread into a single layer and roast until tender, 6-8 minutes.
Make the Sauce
Return pan used to cook chicken to medium-high heat. Add minced garlic and sun-dried tomatoes and cook until lightly browned and aromatic, 60-90 seconds. Add 2 tsp. lemon juice and let evaporate. Add wine and reduce until almost evaporated. Add ½ cup water. Bring to a simmer and cook until it coats the back of the spoon, 5-7 minutes. Remove from burner and swirl in remaining butter. Season with a pinch of salt and pepper.
Finish the Dish
Pile potatoes mousseline on plate. Place chicken alongside and top with sauce. Add green beans to plate.