Start by melting 1 tbsp of your butter in your pot. Add your finely diced onions and sautee them until they are caramelized. This will develop a lot of yummy flavor.
Once your onions are brown and caramelized, add the rest of your butter and your flour to the mixture. Stir and heat until the mixture becomes a paste. You are making a rue right now.
Once the flour is mixed and has cooked for a couple minutes, add your half and half. Use a whisk to mix it, then add the chicken stock. Keep it on medium heat and let it come to a boil. Continue heating until it boils and starts to thicken.
At this point, add the carrots and broccoli. Let it all cook together until the broccoli and carrots are softened. This takes about 10-15 minutes. Keep an eye on it and don't let it boil over or burn on the bottom. Turn the heat to medium low.
Once the veggies are softened, turn off the heat. Add your cheese and mix until well combined.
At this point you can season as needed. I add 1-2 tbsp dijon mustard and about 1 tsp Tabasco sauce along with the nutmeg and stir to combine. I add a pinch of salt and a pinch of pepper as well. I like to taste it at this point and see if it needs more seasoning. I often will end up adding a bit more of each, but I like to add a little bit at a time in order to prevent over seasoning. I love adding enough Tabasco sauce to give it a kick. My kids love the flavor. It's not hot or spicy, but it has a bite to it.
This is wonderful served with homemade wheat bread or some yummy garlic bread! This is wonderful the next day after it has had time to blend its flavors even more. My kids love taking this for lunch to school.