Step 1: Melt the butter in a large pot or Dutch oven over medium heat. Add the diced onion and cook until softened, about 5-7 minutes. Stir in the minced garlic and cook for another minute until fragrant.
Step 2: Sprinkle the flour over the softened onions and garlic. Cook, stirring constantly, for 1-2 minutes to create a roux. This will thicken your soup.
Step 3: Gradually whisk in the chicken (or vegetable) broth until smooth, ensuring no lumps remain. Bring the mixture to a gentle simmer, stirring occasionally, until it starts to thicken.
Step 4: Pour in the half-and-half (or whole milk) and continue to simmer gently for about 10 minutes, stirring frequently, until the soup is creamy and coats the back of a spoon. Do not boil.
Step 5: Add the finely chopped broccoli florets and grated carrots to the pot. Cook for 10-15 minutes, or until the vegetables are tender-crisp.
Step 6: Reduce the heat to low. Gradually stir in the freshly grated sharp cheddar cheese, a handful at a time, allowing each addition to melt completely before adding more. Stir continuously until the soup is smooth and cheesy. Do not let the soup boil after adding the cheese, as this can cause it to separate.
Step 7: Season with salt, black pepper, and a pinch of nutmeg if desired. Taste and adjust seasonings as needed. Serve hot with crusty bread or croutons.