Sauté the Onions: Melt 1 tablespoon of butter in a skillet over medium-high heat. Add the chopped onion and sauté until it’s translucent, about 5 minutes. Set aside.
Prepare the Base: In a large saucepan over medium-low heat, whisk together the 1/4 cup melted butter and carbquick (or flour). Cook for 3-4 minutes, ensuring you stir constantly to prevent burning. Still whisking, slowly add in the heavy cream/water mixture as needed.
Simmer with Stock: Pour the chicken stock into the thickener mixture while continuing to whisk. Bring it to a simmer and cook for about 20 minutes, or until the mixture thickens.
Add Veggies: Stir in the chopped broccoli, carrots, and the sautéed onions. Simmer until the vegetables are tender, approximately another 20 minutes.
Cheesy Finish: Finally, mix in the shredded cheddar cheese until it melts. Season with salt and pepper to taste. Serve warm and enjoy!