Preheat your oven to 425°F (220°C) and line a large baking sheet with parchment paper or a silicone mat for easy cleanup.
Pat the chicken thighs dry with paper towels. This helps the BBQ glaze stick and the skin crisp. In a small bowl, mix olive oil, minced garlic, paprika, salt, and pepper. Rub half of this mixture over the chicken thighs and place them on the prepared sheet pan, skin side up.
Toss the sweet potato cubes with the remaining olive oil and a pinch of salt. Scatter them around the chicken on the sheet pan so there’s a single layer for even roasting.
Roast for about 15 minutes, then remove the sheet and brush the chicken with BBQ sauce. Toss the broccoli with a quick drizzle of oil and a pinch of salt, then add it to the pan alongside the potatoes.
Return the tray to the oven and roast for another 12–15 minutes, or until the chicken reaches an internal temperature of 165°F (74°C) and the potatoes are tender with a light caramelization around the edges.
If you like a bit more glaze, brush the chicken with a little more BBQ sauce and broil for 1–2 minutes, watching carefully to prevent burning.
Remove from the oven and let rest for 5 minutes. Spoon any extra pan juices over the chicken and vegetables, then serve hot.