In a large pot, melt the butter over medium heat. Add the chopped onion and cook until softened, about 5 minutes.

Stir in the minced garlic and cook for another minute until fragrant. Sprinkle in the flour and cook, stirring constantly, for 2 minutes to create a roux.

Gradually whisk in the chicken or vegetable broth until smooth, then bring the mixture to a simmer, stirring occasionally.

Add the broccoli florets to the pot and cook for 10-15 minutes, or until the broccoli is tender. Use an immersion blender to partially blend the soup, leaving some small chunks of broccoli for texture.

Reduce the heat to low and stir in the milk, salt, and black pepper. Gradually add the shredded cheddar cheese, stirring until it is completely melted and the soup is smooth and creamy. Do not boil the soup after adding the cheese to prevent it from separating.

Taste and adjust seasonings as needed. Serve hot, optionally garnished with extra shredded cheese or croutons.