Preheat your oven to 400°F (200°C) and line a large rimmed baking sheet with parchment paper for easy cleanup.
Slice the sweet potato into uniform 1/2-inch thick wedges and place them on one side of the prepared baking sheet.
Drizzle the sweet potatoes with 0.5 tbsp of olive oil and a pinch of salt and pepper, tossing to coat evenly.
Place the baking sheet in the oven and roast the potatoes for 10 minutes to give them a head start.
While the potatoes roast, combine the smoked paprika, garlic powder, onion powder, salt, and pepper in a small bowl.
Pat the chicken breast dry with a paper towel, rub with the remaining 0.5 tbsp of olive oil, and coat both sides thoroughly with the spice blend.
Remove the sheet from the oven, move the potatoes to one side, and add the chicken and broccoli florets to the empty space.
Return the sheet to the oven and roast for an additional 18-22 minutes, or until the chicken reaches an internal temperature of 165°F and the potatoes are golden brown.
Let the chicken rest for 5 minutes before slicing to ensure the juices stay locked in.