Preheat the oven to 425°F. Pat the chicken dry with paper towels. Sprinkle the chicken on both sides with salt and pepper and place on a lightly greased large rimmed baking sheet.
Whisk together 2 tablespoons mustard and 1½ tablespoons honey, the thyme, vinegar, and garlic powder in a small bowl. Brush the mixture evenly over the chicken. Bake for 10 minutes.
Meanwhile, peel the butternut squash and cut in half. Scoop out the seeds. Cube the butternut squash. Trim and cut the broccoli into spears. Peel and quarter the shallots.
Toss the vegetables with the oil, 1 tablespoon mustard, 1 tablespoon honey, salt, and pepper in a large bowl.
Add the vegetables to pan of chicken, nestling as much into a single layer as possible. Bake for 25 more minutes, or until the chicken registers 165°F when a thermometer is inserted in the thickest portion and the vegetables are browned and tender, stirring occasionally to allow the vegetables to caramelize.