Grate the cheese and let it come to room temperature.
Melt the butter in a large pot or dutch oven, over medium heat.
Add the onions and heat until soft and translucent, about 5 minutes.
Add the garlic and cook for an additional minute.
Add one Tablespoon of flour and stir to combine. Heat for one minute, stirring continuously.
Slowly pour in the chicken broth and stir to combine. Add in the nutmeg.
Add the half and half, increase the heat slightly to bring the mixture to a light bubble and allow it to thicken, about 5 minutes.
Reduce the heat to medium. Add the carrots and broccoli to the soup and let the mixture simmer until the vegetables are very tender. Stir occasionally as it cooks.
Turn off the heat and place the pot on a cool burner. Allow the base to cool for 5 minutes.
Sprinkle the remaining 1 Tablespoon of flour over the shredded cheese.
Add the cheese/flour in 4 separate batches, stirring to incorporate as each batch is added.
Once the cheese is melted and combined, serve it immediately and enjoy!
Instead of half and half, you can use one cup of heavy cream and one cup of milk.
Alternatively, you can use all milk, but note that the soup won't be as thick and creamy.