Preheat your oven to 400°F (200°C) and line a large rimmed baking sheet with parchment paper.
Peel the sweet potato and cut it into 1-inch cubes, then slice the carrots into half-inch rounds.
Place the chicken breast, sweet potato cubes, and carrot rounds onto the prepared baking sheet.
In a small bowl, mince the garlic and whisk it together with the olive oil, rosemary, thyme, salt, and pepper.
Drizzle the herb oil over the chicken and vegetables, using your hands or tongs to toss everything until thoroughly and evenly coated.
Roast in the center of the oven for 15 minutes.
Remove the pan briefly to add the broccoli florets, tossing them in the pan juices to coat.
Return the pan to the oven and roast for an additional 10-15 minutes until the chicken is cooked through to 165°F and the vegetables are golden and tender.