Preheat Oven to 425°F. If you want maximum browning, put your Baking Sheet in the oven while it preheats. Cut broccoli — stalk included — into coins and florets roughly 1½–2 inches across; uniform sizing means even cooking. Line the hot pan with Aluminum Foil or Parchment Paper. Toss broccoli with about 2 tbsp olive oil per 1–1½ lbs, season well with kosher salt and black pepper, and add smashed garlic if using. Spread in a single layer with no pieces touching — crowding steams instead of roasts.

Place the Baking Sheet on the center rack of your preheated 425°F Oven. If the pan was preheated, transfer the broccoli carefully onto the hot surface to jump-start browning. Roast 18–24 minutes. At the 10-minute mark, check the undersides. Flip or toss with a metal Spatula for even browning, or leave undisturbed for one deeply caramelized side. Look for deep golden-brown edges, crispy floret tips, and stems that are crisp-tender when pierced with a fork. Do not use convection — it over-dries the broccoli. If edges are browned but stems are still firm, pull the pan, cover tightly with Aluminum Foil, and rest 5–10 minutes; the steam will finish the stems without losing the browning.

Add the 4 cups broccoli florets cut into small pieces, about 1 large head, 4 cups chicken broth low-sodium, ½ tsp smoked paprika, ¾ tsp kosher salt plus more, and ½ tsp black pepper freshly ground. Stir to combine. Bring to a boil, then reduce heat to medium-low. Simmer uncovered for 15–18 minutes, until the 4 cups broccoli florets cut into small pieces, about 1 large head is very tender and easily pierced with a fork.

Use an Immersion Blender to partially blend the soup — pulse a few times to break down most of the 4 cups broccoli florets cut into small pieces, about 1 large head while leaving some small chunks for texture. Alternatively, transfer about half the soup to a Blender, blend until smooth, and stir it back into the Pot.

Reduce heat to low. Stir in the 1 cup heavy cream and 1 tsp Dijon mustard. Let the soup warm through for 2–3 minutes — do not boil at this stage.

Remove the Pot from heat. Add the shredded 3 cups sharp cheddar cheese freshly shredded, packed in two or three handfuls, stirring well between each addition until fully melted and smooth. Taste and adjust ¾ tsp kosher salt plus more and ½ tsp black pepper freshly ground as needed.