Using vegetable peeler, peel outside of broccoli stems and trim ends. Cut stems from broccoli head and finely chop to yield 1 cup. Cut florets into small pieces to yield 5 cups.

In large Dutch oven or heavy-bottomed pot, melt 3 tbsp of the butter over medium heat. Cook onion, stirring, until softened, 5 to 7 minutes. Add garlic and cook for 1 minute.

Add broccoli stems, ½ cup water, ½ tsp each of the salt and pepper and cook, stirring, until tender, 8 minutes. Transfer to bowl. Add remaining 3 tbsp butter to pot over medium heat; add flour and cook, stirring constantly, until golden, 1 minute.

Pour in chicken broth in slow, steady stream, whisking constantly until smooth. Add milk, potatoes, reserved broccoli and remaining 1 tsp salt and ½ tsp pepper. Bring to a boil over high heat. Reduce heat and simmer over low, stirring occasionally, until vegetables are very tender, 25 to 30 minutes.

Stir in cheddar. Working in batches, purée soup in blender until smooth. Pour into clean Dutch oven; heat over medium-low. Add hot sauce.

To serve, divide soup among bowls; top with remaining cheese.