Preheat oven to 475°F (245°C). Position rack in upper-middle of oven.
Mix vegetables: In a large bowl, toss Brussels sprouts, potatoes, carrots, shallots, and garlic with olive oil, 2 tsp thyme, 1 tsp rosemary, sugar, ¾ tsp salt, and ¼ tsp pepper.
Season chicken: In a small bowl, mix melted butter with remaining thyme and rosemary, ¼ tsp salt, and a pinch of pepper. Pat chicken dry and season with salt and pepper.
Assemble: Spread vegetables in a single layer on a large rimmed baking sheet. Place Brussels sprouts in the center. Lay chicken thighs skin-side up on top.
Brush chicken with herb butter.
Roast for 35–40 minutes, rotating the pan halfway through, until the chicken reaches 175°F internally.
Rest 5 minutes, then serve hot. Toss vegetables in pan juices before serving.