Prepare vegetables by grating or chopping. Shred cheese and set aside to allow it to come to room temperature.
Add vegetable broth to a large soup pot over medium heat. Add the carrots, celery and potatoes, cover pot, and bring to a boil. Once boiling, reduce heat and simmer until the vegetables are tender, 3-5 minutes.
Add chopped broccoli and simmer for another minutes or two, depending on how chunky or fine you chopped it. Remove from heat.
In a separate skillet over medium heat, sauté onion in 1 tablespoon butter until tender and lightly browned.
Add onion to pot with vegetables. Add remaining 4 tablespoons of butter to the skillet, over medium heat. Once melted, whisk in ¼ c. flour and stir well to make a roux.
Add the half and half and milk, one cup at a time, stirring constantly. Cook roux, stirring continually with whisk until thickened, smooth and bubbly. Season well with salt and pepper.
Slowly add rue to the pot of cooked vegetables, stirring carefully to combine. Cook for 2 minutes.
Remove from heat and add grated cheese, stirring until melted. Taste and season with additional salt and pepper or vegetable bouillon base, to taste, if needed.
To make it Healthier: use a leaner milk and half and half.Instant Pot: Add broth and all veggies to the instant pot. Cook on high pressure/manual for 2 minutes. Quick release steam. Cook roux in a separate skillet and add to the pot. Stir in cheddar cheese. Slow Cooker: Add broth and all vegetables to the slow cooker. Cook on HIGH for 4 hours or LOW for 6-8 hours. Cook roux in a separate skillet and add to the pot. Stir in shredded cheddar cheese. Storing and Freezing: This broccoli cheese soup will keep well in the fridge for 3-4 days. You can freeze it before adding the roux, milk and cream (otherwise the half and half may separate upon reheating and the soup could develop a slightly grainy texture.)