Basic + Awesome Broccoli Cheese Soup

This Broccoli Cheese Soup is easy, creamy, and a perfect cozy classic. A silky-smooth soup base, little bites of just-right broccoli and carrots, and some sharp creamy cheddar cheese to finish it off.

• 5 tablespoons butter, divided
• 1 small onion, minced
• 3 cloves garlic, minced
• 1 cup broccoli stems, chopped (optional)
• 1/4 cup all-purpose flour
• 1 cup whole milk
• 1 cup half and half
• 2 1/2 cups chicken or vegetable broth
• 2–3 cups broccoli florets, chopped into very small pieces
• 1 large carrot, thinly sliced
• 1 teaspoon salt (more or less to taste)
• 1/2 teaspoon freshly ground black pepper
• 1/2 teaspoon paprika
• an 8-ounce block of good quality extra-sharp cheddar cheese, grated