Prepare the broccoli: Cut the broccoli into small florets. You can also peel and chop the broccoli stems if desired.

Cook the onion and garlic: In a large pot or Dutch oven, melt the butter over medium heat. Add the chopped onion and minced garlic. Cook, stirring occasionally, until the onion is translucent and the garlic is fragrant, about 3-4 minutes.

Make the roux: Sprinkle the flour over the onion and garlic mixture. Stir well to combine and cook for another 1-2 minutes to cook out the raw flour taste, stirring constantly.

Add broth and broccoli: Gradually pour in the chicken or vegetable broth, stirring constantly to prevent lumps. Bring the mixture to a simmer.

Simmer the soup: Add the broccoli florets to the pot. Simmer gently for about 10-15 minutes, or until the broccoli is tender but still bright green.

Blend the soup (optional): For a smoother consistency, you can blend the soup using an immersion blender directly in the pot, or transfer portions to a blender and blend until smooth. Be cautious with hot liquids in a blender.

Add cream and cheese: Stir in the half-and-half or heavy cream until well combined. Gradually add the shredded cheddar cheese, stirring continuously, until the cheese is melted and the soup is creamy.

Season: Season the soup with salt and pepper to taste. Adjust seasoning as needed.

Serve: Ladle the soup into bowls and garnish with croutons, extra shredded cheese, or chopped fresh herbs if desired.

Enjoy: Serve hot and enjoy the comforting flavors of homemade Broccoli Cheese Soup!