In a 5-quart enameled Dutch oven or heavy soup pot, melt 4 tablespoons of butter over medium heat. Add the finely diced yellow onion and sauté for 4 to 5 minutes until soft and translucent. Stir in the minced garlic and cook for 1 minute until fragrant.

Sprinkle 1/4 cup of all-purpose flour evenly over the sautéed onions. Whisk constantly for 1 to 2 minutes to cook out the raw flour taste and form a light golden bubbling roux paste.

Slowly stream in the low-sodium broth, room-temperature whole milk, and heavy cream while whisking continuously to ensure a silky-smooth, lump-free base. Whisk in the Dijon mustard, ground nutmeg, sea salt, and black pepper.

Bring the broth to a gentle simmer over medium heat. Add the chopped broccoli florets and matchstick carrots. Reduce the heat to medium-low, cover partially, and let simmer gently for 12 to 14 minutes, stirring occasionally, until the broccoli and carrots are tender-crisp.

Remove the pot completely from the heat source (or turn heat to dead-low). Add the freshly grated sharp cheddar cheese one generous handful at a time, whisking gently in circles until completely melted and glossy before adding the next handful. Never boil after cheese is added!

Let the soup rest off the heat for 2 minutes to thicken into a velvety, restaurant-style consistency. Ladle into warm rustic ceramic bowls, garnish with extra freshly grated cheddar, cracked black pepper, and chopped fresh parsley. Serve immediately with warm crusty sourdough rolls.