In a large pot, melt the butter over medium heat. Add the diced onion and sauté for about 5 minutes, or until translucent. This step builds the flavor base of the soup.

Add the minced garlic and sauté for an additional 1 minute, until fragrant. Be careful not to burn the garlic.

Sprinkle the flour over the onion and garlic mixture, stirring constantly for about 2 minutes to create a roux. This will help thicken the soup.

Gradually whisk in the chicken broth, ensuring there are no lumps from the roux. Bring the mixture to a simmer.

Once the broth is simmering, add the broccoli florets and shredded carrot. Cook for about 10-15 minutes, or until the broccoli is tender.

Reduce the heat to low and stir in the heavy cream, salt, black pepper, and cayenne pepper (if using). Allow the soup to warm through for about 5 minutes.

Gradually add the shredded cheeses, stirring until melted and smooth. This creates the creamy, cheesy texture characteristic of broccoli cheddar soup.

Ladle the soup into bowls and serve hot. Optionally, garnish with additional cheese or fresh cracked pepper.