Preheat your oven to 425°F (220°C). Line a large baking sheet with parchment paper or a silicone mat for easy cleanup.
In a big bowl, toss the diced sweet potatoes with 1.5 tablespoons of olive oil, half the minced garlic, paprika, thyme, salt, and pepper. Spread them in a single layer on the sheet pan.
Roast the potatoes for about 10 minutes. While they start to soften, cut the broccoli into small florets and slice the chicken sausage into coins.
After the potatoes have roasted briefly, remove the pan from the oven. Add the broccoli and sausage coins. Drizzle with the remaining 1.5 tablespoons olive oil, the rest of the garlic, and a squeeze of lemon juice if using.
Toss everything together gently to coat, then spread in an even layer. Return to the oven and roast for another 12–15 minutes, until the sausage is heated through and the broccoli is vibrant and crisp-tender.
If you want extra color and a bit more char, switch to the broiler for 1–2 minutes at the end, but watch closely to prevent burning.
Remove from the oven and let rest for a couple of minutes before serving. A light squeeze of lemon or a sprinkle of fresh parsley can brighten the dish just before serving.