Melt the butter in a deep pot over medium heat, then cook the chopped onion and minced garlic for 1 to 2 minutes, stirring often. The onion should begin to soften and the garlic should smell fragrant without browning.

Pour in the chicken broth and heavy cream, then add the finely chopped broccoli, smoked paprika, pepper, and salt; bring the mixture to a boil before lowering the heat and simmering for 20 minutes. The broccoli is ready when the pieces are fork-tender and the broth looks lightly thickened.

Take the pot off the active heat and stir in the cheddar 1/2 cup at a time until each addition has melted before adding the next. The finished soup should look glossy and creamy, with no strings of unmelted cheese; serve with extra shredded cheese and chopped parsley on top.