Preheat the oven to 400°F, and line a large baking sheet lined with parchment paper.
Arrange the potatoes, carrots and onions on the baking dish, drizzle 1 tablespoon olive oil, season with 1 teaspoon salt and ½ teaspoon black pepper and toss to coat. Roast the vegetables in the preheated oven for 25 minutes, tossing halfway through.
In another small baking sheet lined with parchment paper, add the broccoli, drizzle with remaining olive oil, and season with remaining salt and pepper; toss to combine. Roast the broccoli for 10 minutes.
Transfer all the vegetables, except for the broccoli to a deep pot. Season with paprika and mustard powder and toss to combine. Pour the vegetable broth over the roasted vegetables and bring to a boil.
Lower heat to a simmer, and blend using an immersion blender until smooth and creamy. Add the broccoli and pulse a few times to incorporate the broccoli. Stir in cheddar cheese and milk.
Serve warm with extra cheese on top, if desired.