Melt the butter in a large Dutch oven or soup pot over medium heat.
Add the onion, carrots, and celery. Cook for 5 to 6 minutes, stirring occasionally, until the vegetables begin to soften.
Add the garlic, dried thyme, salt, black pepper, and smoked paprika. Stir for 30 seconds, until fragrant.
Sprinkle the flour over the vegetables. Stir well and cook for 1 minute to remove the raw flour taste.
Slowly pour in the vegetable broth while stirring continuously. Add the diced potatoes and bring the mixture to a gentle boil.
Reduce the heat to a simmer. Cook for 12 to 15 minutes, partially covered, until the potatoes are nearly fork-tender.
Add the broccoli florets and corn. Simmer for 5 to 7 minutes, until the broccoli is tender.
Stir in the whole milk and heavy cream. Warm gently over low heat. Do not let the chowder boil after adding dairy.
Remove the pot from the heat. Add the freshly grated cheddar a handful at a time, stirring after each addition until fully melted and smooth.
Taste and adjust salt and pepper. For a thicker chowder, mash a few cooked potato pieces against the side of the pot.
Serve hot with your preferred toppings, such as extra cheddar, parsley, crackers, or bacon.