Place a 5- to 6-quart Dutch oven or heavy-bottomed pot over medium heat. Add the 4 tablespoons butter and let it melt.

Add the diced onion and carrots: Cook for 6–8 minutes, stirring occasionally, until the onion is translucent and the carrots begin to soften. They should not be deeply browned.

Stir in the 3 minced garlic cloves and cook for about 30 seconds, just until fragrant. Keep the heat at medium so the garlic does not scorch.

Sprinkle the 1/4 cup flour over the vegetables and stir continuously for 1–2 minutes. The flour should coat the vegetables and form a smooth, pale paste without developing a dark color.

Slowly whisk in the 3 cups chicken broth, followed by the 2 cups whole milk. Add the liquids gradually while whisking to prevent lumps. Continue cooking over medium heat until the mixture begins to steam and lightly thicken, about 5 minutes.

Add the 5 cups broccoli florets: Reduce the heat to medium-low and simmer gently for 12–15 minutes, stirring occasionally. The broccoli should be fork-tender but still hold its shape. Avoid a hard boil.

Stir in the 1 cup heavy cream and warm for about 2–3 minutes. Do not let the soup come to a vigorous boil.

Reduce the heat to low: Add the 3 cups shredded sharp cheddar a handful at a time, stirring after each addition until melted before adding more. The soup should become smooth, thick, and creamy.

Stir in the 1 teaspoon kosher salt and 1/2 teaspoon black pepper. Taste and adjust the seasoning if needed. Ladle into bowls and top with additional shredded cheddar or fresh parsley, if using.