Melt unsalted butter in a large Dutch oven or heavy-bottomed pot over medium heat. Add the finely diced yellow onion and cook for 4 to 5 minutes until soft, tender, and translucent. Stir in minced garlic and cook for 1 minute until fragrant.

Sprinkle all-purpose flour over the sautéed onions and melted butter. Whisk continuously for 1 to 2 minutes to cook out raw flour flavor and create a smooth golden roux.

Slowly pour in the chicken broth in a steady stream while whisking constantly to ensure no lumps form. Pour in the half-and-half while continuing to whisk. Bring the mixture to a gentle simmer over medium heat.

Stir in the chopped broccoli florets and grated carrots. Season with ground nutmeg, sea salt, and black pepper. Reduce heat to medium-low and simmer uncovered for 12 to 15 minutes, stirring occasionally, until the broccoli is tender and easily pierced with a fork.

Remove the pot from direct heat (or set to absolute lowest heat). Add freshly shredded sharp cheddar cheese one generous handful at a time, stirring gently between additions until completely melted and silky smooth. Do not boil once cheese is added.

Taste and adjust salt and black pepper as desired. Ladle into warm bowls, top with extra shredded cheddar, and serve immediately with warm crusty sourdough bread.