Melt the butter in a large Dutch oven over medium heat. Add the finely diced onion and sauté for 4 minutes until soft and translucent. Stir in the minced garlic and cook for 1 minute.

Sprinkle the flour, dry mustard powder, and nutmeg over the onions. Whisk constantly for 90 seconds until golden and bubbly.

Slowly pour in the broth while whisking continuously until smooth. Stir in the chopped broccoli florets and julienned carrots.

Bring to a gentle simmer, reduce heat to medium-low, cover, and cook for 10-12 minutes until the broccoli and carrots are tender.

Turn the heat to low and stir in the warm half-and-half (or heavy cream). Remove the pot from the burner and add the freshly grated sharp cheddar cheese one handful at a time, stirring until velvety smooth. Serve warm in bowls or sourdough bread bowls.