Melt 1 tablespoon of butter in a large Dutch oven or heavy soup pot over medium heat. Add diced onions and cook for 3 to 4 minutes until translucent. Add minced garlic and cook for 1 minute until fragrant. Transfer onions and garlic to a small dish.

In the same pot, melt the remaining 3 tablespoons of butter over medium-low heat. Whisk in the flour and cook for 1 to 2 minutes until bubbly and pale golden to cook out raw flour taste.

Slowly pour in chicken broth and half-and-half in a steady stream, whisking constantly to prevent lumps. Whisk in nutmeg, dry mustard powder, salt, and black pepper.

Add the cooked onions, chopped broccoli florets, and matchstick carrots. Bring to a gentle boil, then reduce heat to low and simmer uncovered for 12 to 15 minutes, stirring occasionally, until broccoli is fork-tender.

Turn the burner heat to LOW (or remove from heat). Gradually fold in the freshly grated sharp cheddar and Monterey Jack cheese, one handful at a time, stirring until completely melted, velvety, and silky smooth.

Taste and adjust salt or black pepper as needed. Ladle hot soup into warmed bowls or hollowed sourdough bread bowls, top with extra shredded cheddar, and serve immediately!