Preheat your oven to 400°F (200°C) and line a large baking sheet with parchment paper.
Peel and cube the sweet potato into 1-inch pieces and slice the carrots into rounds.
In a large bowl, toss the sweet potatoes and carrots with half of the olive oil, half of the salt, and half of the pepper.
Spread the potatoes and carrots onto the baking sheet and roast for 10 minutes.
While the vegetables roast, mince the garlic and whisk it together with the remaining olive oil, lemon juice, rosemary, and thyme in a small bowl.
Brush the chicken breast with the lemon-herb mixture and season with the remaining salt and pepper.
Remove the baking sheet from the oven, add the chicken breast and broccoli florets to the tray, and return to the oven.
Roast for an additional 15-20 minutes, or until the chicken reaches an internal temperature of 165°F and the vegetables are tender and golden.