In a large pot, melt the unsalted butter over medium heat. Add the diced onions and sauté until they are translucent and fragrant, about 5 minutes.
Sprinkle the flour over the softened onions and stir continuously for about 2 minutes to cook out the raw flour taste.
Gradually whisk in the heavy cream (or half-and-half), stirring to combine. Continue to cook, allowing the mixture to thicken slightly.
Slowly add the vegetable broth, starting with 2 cups. Increase the amount based on your preferred soup thickness. Stir to combine.
Add the chopped broccoli and julienned carrots to the pot. Bring the mixture to a gentle simmer and let it cook for about 15-20 minutes, or until the vegetables are tender.
Stir in the nutmeg and grated sharp cheddar cheese, allowing the cheese to melt completely. Season with kosher salt and freshly ground black pepper to taste.
Ladle the soup into bowls and serve hot. Pair it with a slice of crusty bread or a fresh salad for a hearty meal!
