Melt the unsalted butter in a heavy-bottomed Dutch oven over medium heat. Add the diced yellow onion and sauté for 3-4 minutes until translucent. Stir in the minced garlic and cook for 30 seconds.

Sprinkle the all-purpose flour evenly over the butter and onions. Whisk vigorously for 1 to 2 minutes until it resembles wet sand and smells slightly nutty.

Slowly pour in the chicken broth while whisking constantly to prevent lumps. Immediately whisk in the half-and-half until the liquid is entirely smooth. Stir in the dry mustard powder and paprika.

Bring the pot to a gentle simmer over medium-low heat. Add the bite-sized broccoli florets and julienned orange carrots. Cover and simmer for 15-20 minutes, until the broccoli is tender but still bright green.

Remove the Dutch oven from the heat entirely. Add the freshly grated sharp yellow cheddar cheese one small handful at a time, stirring gently in a figure-eight motion until melted before adding more.

Use a serrated knife to slice the top quarter off each sourdough loaf. Hollow out the soft interior bread, leaving a 1-inch thick crust wall. Ladle the hot soup into the bread bowls and garnish with a pinch of extra grated cheddar.