In a large pot, melt 3 tablespoons of butter over medium heat. Once melted, add 1 cup of chopped carrots, 1 cup of chopped celery, 1 cup of chopped onion, and 2 minced garlic cloves. Sauté for about 5-7 minutes, until the veggies are tender and the onion is translucent.

Stir in 1/4 cup of all-purpose flour and cook for an additional 1-2 minutes, stirring continuously. Gradually add in 4 cups of low-sodium chicken broth while whisking to avoid lumps. Continue stirring until the mixture thickens and begins to bubble gently, roughly 3-4 minutes.

Introduce 4 cups of peeled and cubed russet potatoes and 1 teaspoon of dried thyme into the pot. Stir well and reduce heat to medium-low, allowing the soup to simmer for about 15 minutes or until the potatoes are fork-tender, stirring occasionally.

Add 2 cups of chopped broccoli florets to the pot and cook for an additional 5 minutes. The broccoli should turn bright green and become tender but not mushy.

Lower the heat and stir in 1 cup of milk and 1/2 cup of heavy cream. Mix until combined and gently heat through for about 3-4 minutes; avoid boiling.

Remove the pot from heat, and fold in 1.5 cups of shredded sharp cheddar cheese and 1/2 cup of finely shredded parmesan cheese. Stir until melted and well combined.

Taste the soup and season with salt and black pepper. Ladle the Cheddar Broccoli Potato Soup into bowls and serve warm, perhaps alongside thick slices of crusty bread.