4ouncesextra sharp cheddar cheeseshredded (about 1 cup), plus additional for serving, do not use pre-shredded cheese, which does not melt as smoothly
1/4cupfinely grated Parmesan or 3 tablespoons nutritional yeast
In a large Dutch oven or similar large, sturdy soup pot, melt the butter over medium heat. Add the onion, salt, and black pepper. Cook until the onion softens and begins to turn translucent, adjusting the heat as needed so it becomes tender but does not brown, about 5 to 8 minutes.
Add the carrot, broccoli, cauliflower, and garlic. Cook until the broccoli is very lightly sautéed and bright green, about 5 minutes.
Add the broth. Bring to a simmer over medium-high.
Reduce the heat to maintain a gentle simmer and cook, stirring periodically, until the vegetables are completely tender, about 15 minutes.
Puree the soup, either in batches in a blender (be careful not to overfill it—blenders love to splatter) or directly in the pot with an immersion blender. Leave the soup as chunky or as tender as you like.
Heat the pot over low. Stir in the milk (or half-and-half), mustard, nutmeg, and cayenne.
Place the cheddar in a bowl and sprinkle the cornstarch over the top. With your fingers, toss to coat the shreds evenly.
Working only 1 small handful at a time, add the cheese very slowly to the soup, stirring constantly and allowing the cheese to melt completely before the next addition. Stir in the Parmesan. Taste and adjust the seasoning as desired. Enjoy warm with crusty bread and additional cheese sprinkled on top.
TO STORE: Refrigerate soup in an airtight storage container for up to 5 days.
TO REHEAT: Rewarm leftovers in a pot on the stovetop over medium-low heat or in the microwave.
TO FREEZE: Freeze soup in an airtight, freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.