Peel and slice shallot into ¼" rounds. Mince two rounds. Place shallot rounds in a mixing bowl with vinegar (reserve ¼ tsp. for sauce) and a pinch of salt. Marinate at least 10 minutes. Stem parsley. Mince stems and coarsely chops leaves. Trim ends off beet, peel, and cut into ¾" dice.
Cook the Beets
Place a medium non-stick pan over medium-high heat. Add ½ Tbsp. olive oil, beets, and a pinch of salt to hot pan. Stir occasionally, 2-3 minutes. Add ¼ cup water and cover. Cook until beets are fork-tender, 2-3 minutes. Add more water as needed. While beets are cooking, drain marinated shallots, reserving liquid. Transfer beets to a plate. Wipe pan clean and reserve.
Cook the Butternut
Return pan used to cook beets to medium-high heat and add ½ tsp. olive oIl. Add butternut squash and a pinch of salt to hot pan. Stir a few times to coat with oil, 2-3 minutes. Add shallot pickling liquid and reduce to almost evaporated. Add ¼ cup water, cover, and cook until tender, 7-10 minutes. Transfer squash to a plate. Wipe pan clean and reserve.
Cook the Chicken
Return pan used to cook butternut squash to medium-high heat. Pat chicken breasts dry, and season with a pinch of salt and pepper. Add 1½ Tbsp. olive oil and chicken to hot pan. Reduce heat to medium and cook until browned, 5-7 minutes. Flip, and cook until chicken reaches a minimum internal temperature of 165 degrees, 6-8 minutes. Transfer chicken to a plate. Reserve pan; no need to wipe clean.
Make Butter and Finish Dish
In a mixing bowl, whisk ¼ tsp. shallot, ¼ tsp. parsley stems, ¼ tsp. vinegar, butter, goat cheese (reserve a pinch for garnish) vigerously until smooth like cake frosting. The more it's whisked, the creamier it will be.Return pan used to cook chicken to medium-low heat. Add remaining shallot, remaining parsley stems, ½ tsp. olive oil, and a pinch of salt to hot pan and cook until tender, 1-2 minutes. Place chicken on a plate and alternate butternut squash and beets next to them. Garnish dish with cooked minced shallot, pickled shallot, parsley leaves, and whipped butter.