Chop white onion, carrot, celery, and set aside.
Mince garlic and set aside.
Peel potatoes and chop into ½ inch cubes. Set aside.
Remove leaves from cauliflower and chop into florets. Set aside.
Cut broccoli florets and a few of the stems. Set aside.
Chop green onion and set aside.
In Dutch oven or large pot, heat over medium heat and add butter.
Add diced onion, carrot, celery and sauté for 5 minutes.
Add garlic and sauté for 1 minute.
Add flour slowly and whisk into vegetables, Cook for 2 minutes stirring frequently. Add a few tablespoons of vegetable broth if needed to help break up the clumps.
Add vegetable broth and milk then whisk until smooth.
Add pepper, salt, mustard powder, and cayenne pepper then mix well. Let the roux sit for 5 minutes to thicken.
Add broccoli, cauliflower, potatoes, and green onion then bring to a boil. Cover and cook for about 10-15 minutes until all vegetables are fork tender.
Smash veggies with large wooden smooth to break up a bit until you reach desired consistency.
Remove from heat.
Garnish with shredded cheese and parsley.