Step 1

Separate broccoli into florets and stems. Cut florets into bite-size pieces and set aside. Roughly chop stems and reserve separately.

Step 2

Heat oil in a large Dutch oven over high heat until shimmering. Add broccoli florets and cook, without moving, until charred on the bottom, about 1 minute. Stir, season with salt and pepper, and continue cooking, stirring occasionally, until tender and charred on several surfaces, about 1 minute longer. Transfer to a rimmed baking sheet to cool.

Step 3

Return Dutch oven to medium heat and add butter, onion, carrot, and broccoli stems. Season with salt and pepper and cook, stirring frequently, until tender but not browned, about 5 minutes, lowering heat if necessary. Add garlic and cook, stirring, until fragrant, about 30 seconds.

Step 4

Add water or chicken stock, milk, and potato and bring to a boil over high heat. Reduce to a bare simmer and cook, stirring occasionally, until broccoli and potato are completely tender, about 30 minutes.

Step 5

In a large bowl, toss both cheeses together along with mustard powder. Using an immersion blender or working in batches with a countertop blender, blend soup, adding cheese a handful at a time, until completely smooth. Stir in hot sauce and season to taste with salt and pepper. Stir in reserved broccoli florets and pulse with blender a few more times until a few pieces are broken down, but most bite-size pieces remain. Serve immediately.