For the Soup- Chop onions, cauliflower, potato and celery, mince garlic, and shred the carrots. Set cauliflower and potato to the side in a separate bowl.
In a pot, combine 1/4 cup of butter, celery sticks, shredded carrots, onion and garlic.
Sauté vegetables until translucent. You may add some seasoning in this step if you wish.
Add chicken stock and chopped potato and cauliflower. Bring to a boil and cover with a lid.
Boil until vegetables are is soft.
For the Roux- In a pan, melt butter and add flour
Mix butter and flour to create a paste
Optional: Add a ladle's worth of broth to the pan
Add the milk to the pan and turn the heat down to low-medium
Continuously stir the milk mixture in the pan until it has bubbled and thickened. Be careful not to scorch the roux!
Once it is very thick, pour it into the pot with the rest of the vegetable soup
Use an immersion blender and carefully blend all ingredients together
Bring soup to a low boil for 5-10 minutes
Remove from heat and let thicken for a short while
Optional: Add some cheddar cheese to the soup