Trim the stems off the broccoli heads and chop into small florets.
In a large pot over medium heat, combine the butter and onion and sauté until translucent. Add the flour and stir until the mixture lightly browns. Stir in the half-and-half and mix until the contents reach a simmer. Turn the heat to low and add the stock. Simmer for 5-10 minutes.
Mix in the broccoli, carrots, salt, pepper, and nutmeg. Simmer for 10-15 minutes. Add the cheese and stir until it melts and combines with the other ingredients.