Place a large pot over medium heat. Melt 1 tablespoon butter. Add onion and saute for 5 minutes until softened, stirring often. Transfer to a bowl and set aside.
In the same pot, melt 1/4 cup butter. Add flour and whisk to make a roux. Cook for 1-2 minutes. Slowly whisk in chicken broth and then heavy cream until smooth. Simmer for 20 minutes, stirring occasionally.
Add broccoli, carrots, and cooked onions to the pot. Reduce heat to medium-low. Simmer for another 25 minutes, stirring often.
Add nutmeg, salt, and pepper. Stir to combine. Add cheese and stir until melted. Taste and adjust seasoning as needed.
At this point the soup is ready to serve, but if you prefer, you can use an immersion blender to puree the vegetables into the soup. It's delicious both ways.
Portion soup into bowls and garnish with more shredded cheese on top if desired. Serve immediately.
Notes
I get 2-3 main dish servings or 4-6 side dish servings from this recipe.