Sauté the Onions: Sauté the chopped onion in the 1–2 tbsp of butter. Set aside.
Make the Roux: Cook the 1/2 cup melted butter and flour using a whisk over medium heat for 3–5 minutes, or until you see a noticeable golden brown color.
Simmer Stock: Slowly add in the vegetable stock and whisk again to combine. Simmer contents covered (stirring occasionally) for 20 minutes on medium heat.
Combine and Cook: Add the broccoli, carrots, and the sautéed onions into the pot. Stir, then add the half-and-half. Cook covered over low heat for 20–25 minutes. Note: Do not bring to a boil, as this may cause the milk to curdle.
Season and Blend: Add salt, pepper, and nutmeg.
Optional: You can purée half of your soup in a blender or with a handheld immersion blender if you choose, however it isn't a requirement.
Add Cheese: Continue to cook the soup on low heat and slowly add the grated cheese a handful at a time. Stir constantly to avoid clumps.
Serve: Once all the cheese has melted, remove from heat and serve immediately.