Sauté the Aromatics: In a large Dutch oven or heavy-bottomed pot over medium heat, melt 4 tablespoons of butter. Add finely diced onion and cook for 3 to 4 minutes until translucent. Stir in minced garlic and cook for 1 minute until fragrant.

Build the Velvety Roux: Sprinkle flour evenly over onions and garlic. Whisk continuously for 1 to 2 minutes to cook out raw flour taste while lightly browning.

Whisk Broth & Cream: Slowly pour in warm chicken bone broth in a steady stream while whisking vigorously to prevent lumps. Whisk in half-and-half. Bring soup to a gentle simmer over medium heat, whisking frequently as it thickens (about 3 to 4 minutes).

Simmer the Vegetables: Add chopped broccoli florets, shredded carrots, ground nutmeg, dry mustard powder, salt, and pepper. Reduce heat to medium-low, cover pot partially, and simmer gently for 12 to 15 minutes until broccoli is tender and easily pierced with a fork.

Melt Cheese Off Heat: Pull the pot completely off the direct heat source. Add freshly shredded extra-sharp cheddar cheese and grated Parmesan one handful at a time, gently stirring in a figure-eight motion until completely melted, creamy, and glossy.

Texture & Serve: For a thicker texture, use an immersion blender to pulse 3 to 4 times (leaving plenty of whole broccoli florets intact for texture). Ladle steaming soup into warm bread bowls or ceramic bowls, top with extra shredded cheddar, and serve immediately!