Melt 1 TBSP 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 TBSP 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, and sauteed onion; simmer until vegetables are tender, about 15 minutes.
Pulse 2-4 cups of soup in blender and add back to the soup.
Stir cheddar cheese into vegetable mixture until cheese melts.
Season with salt and pepper to taste. Sometimes I add extra onion powder to enhance the flavor.
