Add 2 tablespoons of butter to a medium skillet over medium heat.
Once the butter has melted, add diced onion and sauté for about 3-4 minutes, until translucent. Stir a few times. Once done, remove the skillet from heat and set it aside.
Next, place a medium heavy bottom pot or dutch over medium heat.
Once the pot is warm, add 4 tablespoons of butter and melt. Once melted, add the flour and cook over medium heat for 2-3 minutes. Stir constantly with a whisk. While still whisking, add the chicken broth. Stir to combine.
Add the heavy cream, stir to combine, and simmer for about 10 minutes.
Add the broccoli, carrots, and cooked onion. Add nutmeg. Add salt and pepper to taste. Cook over low heat for 20-25 minutes. Stirring occasionally.
Once done, you can leave the soup chunky or blend about 1/2 of the soup, by adding it to a blender or using an immersion blender. After blessing the soup, stir it in for a smoother soup.
Return to low heat and add the cheddar cheese.
Serve warm with extra cheddar cheese.