I use a mild, aged white cheddar. If you prefer a tangier and strong sharp cheese flavor, use a sharper cheddar.The trick to keeping this thick and creamy vs. watery is to start with less water and add more as necessary. You can also add some water at the end of step 4, after you've added the cheese.
Warm a large saucepan on low-medium heat and then add 1 tablespoon of butter.
Add the diced onion and 1/4 teaspoon salt and cook until just starting to brown, about 8 minutes.
Add the carrots and broccoli to the saucepan, and then add enough water to cover the vegetables and bring them to a boil.
Cover the saucepan and lower the heat to simmer the vegetables for about 12 minutes, or until the broccoli and carrots are tender.
Turn off the heat, take the cover off and add the cheese and salt, and stir until cheese is melted. If you need to, add a bit more water.
Cool for a bit and then blend the soup in batches. Blend part or all of the soup depending on whether you want it creamy or with pieces of broccoli, carrot and onions. If you're finding it too thick, add a tablespoon of water at a time and blend. Salt and pepper to your liking.
Serve warm. Store in the refrigerator for a few days or freeze for a few months.