Melt 1 tablespoon butter in a skillet over medium-high heat. Saute onion in hot butter until translucent, about 5 minutes. Set aside.
Whisk 1/4 cup melted butter and flour together in a large saucepan over medium-low heat; cook until flour loses it's granular texture, adding 1 to 2 tablespoons of milk if necessary to keep the flour from burning, 3 to 4 minutes.
Gradually pour milk into flour mixture while whisking constantly. Stir chicken stock into milk mixture. Bring to a simmer; cook until flour taste is gone and mixture is thickened, about 20 minutes. Add broccoli, carrots, sauteed onion, and celery; simmer until vegetables are tender, about 20 minutes.
Stir Cheddar cheese into vegetable mixture until cheese melts. Season with salt and pepper to taste.