In a medium pot, melt the 1 Tbsp of butter over medium heat. Saute the onions until translucent. Set aside.
Add the 1/4 cup butter to the pot and melt over medium heat. Whisk in the flour and cook for 2-3 minutes. Add half and half and chicken stock. Simmer on low for 10 minutes.
Add broccoli, carrots and onions to the pot. Cook for 20 minutes over low heat. Season with salt and pepper according to taste. Return to heat and stir in cheese.