Start by melting the butter in a large pot over medium heat. Once it’s foamy, add the chopped onions. You want them to become soft and translucent, which should take around 3-5 minutes. Make sure to stir to prevent them from browning. Adding the carrots at this point can create a lovely flavor base.

After the onions and carrots are soft, sprinkle the flour over the mixture. Stir continuously for about 2 minutes to cook the flour. This step is crucial for achieving the right consistency later on. It will help thicken the soup while keeping the flavor intact.

Slowly pour in the chicken broth, whisking as you do so to keep the mixture smooth. The whisk is your best friend here, as it breaks up any clumps formed from the roux. Let the mixture simmer gently for about 10 minutes, allowing the flavors to meld.

Now, it’s time to add the half-and-half. Pour it in slowly, and continue to stir. This addition will make the soup luxuriously creamy. Keep simmering for another 5 minutes while stirring occasionally.

Toss in the broccoli florets and remaining carrots. You want the broccoli to remain vibrant and tender but not mushy. Cook for 5-7 minutes or until the broccoli is a bright green and cooked through.

Finally, stir in the garlic powder, nutmeg, and season with salt and pepper to taste. Add the shredded cheddar gradually, mixing well until it melts into the soup. Then, add the Parmesan cheese for an extra punch.