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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, 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 or directly in the pot with an immersion blender.
Heat the pot over low. Stir in the milk, mustard, nutmeg, and cayenne.
Place the cheddar in a bowl and sprinkle the cornstarch over the top. 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.