Preheat the oven to 425°.
Wash the broccoli and cauliflower and cut the heads into florets. Place them on a baking sheet lined with parchment paper or a silicone mat.
Peel and chop the carrots and place them on the baking sheet with the broccoli and cauliflower.
Peel and chop the onion and place it on the baking sheet with the rest of the vegetables.
Add the olive oil to the vegetables and sprinkle the vegetables with 2 teaspoons of salt. Mix the vegetables around so they are well coated.
Place the baking sheet in the oven and roast for 20 minutes.
After 20 minutes, take the baking sheet out of the oven and spoon the vegetables into a Dutch oven. Cover the vegetables with the broth and bring the mixture to a simmer.
When it begins to simmer, use an immersion blender and carefully blend the soup until it is a creamy consistency.
Once the consistency is to your liking, add the heavy cream and shredded cheddar. Reduce the heat to low and stir until all the cheese is melted.