In a large pot, melt the butter over medium heat. Add the chopped onion and minced garlic, sautéing until the onion becomes translucent, about 5 minutes.

Sprinkle the flour over the onion and garlic mixture, stirring constantly for about 1-2 minutes to create a roux.

Slowly whisk in the vegetable broth, ensuring there are no lumps. Bring the mixture to a simmer.

Add the broccoli florets to the pot and cook until the broccoli is tender, about 10-15 minutes.

Reduce the heat to low and stir in the milk, shredded cheddar cheese, salt, black pepper, paprika, and nutmeg. Stir until the cheese is melted and the soup is creamy.

Serve hot, garnished with additional cheese if desired.