On the stove in a large pot over medium heat, melt the butter and then sauté the onion and garlic until translucent, about 5 minutes.
Once the onions are cooked, sprinkle in the flour and use a whisk to slowly whisk in the half-and-half and stock. Add in the broccoli, carrots, and nutmeg and stir.
Once the broccoli is cooked, stir in the cheese and then taste and adjust the salt and pepper. Use a stick blender to blend the soup a little until still somewhat chunky, but this is your choice.