Preheat oven to 400F and lightly grease a large sheet pan. Toss sweet potatoes, carrots, parsnips, beets, onion, 2 tablespoons olive oil, salt, and pepper to taste together and arrange in a single layer.
Roast vegetables at 400F 20-25 minutes until vegetables are crisp-tender. Remove from oven and keep oven on.
Season chicken on both sides with salt and pepper to taste. Heat 1 tablespoon butter and remaining 1 tablespoon olive oil in a large skillet over medium-high heat until melted and hot. Place chicken thighs skin-side down and sear 3-5 minutes on both sides until golden brown.
Remove chicken from skillet and arrange on sheet pan with root vegetables. Set aside for a moment.
Drain all but 1 tablespoon drippings from pan. Add garlic to pan drippings and sauté over medium heat 1-2 minutes until fragrant. Add apple cider and mustard to pan and bring to a simmer over medium heat. Reduce heat to medium-low and cook until apple cider is reduced in half, about 6-8 minutes.
Whisk remaining 1 tablespoon butter into apple cider sauce until melted. Season sauce with salt and pepper to taste. Drizzle half of sauce over chicken thighs and root vegetables and sprinkle rosemary over top.
Roast chicken and vegetables at 400F 15-20 minutes until chicken registers 165F and vegetables are tender. Drizzle remaining apple cider sauce over top. Serve chicken and vegetables warm and enjoy!