Dice onion and chop cauliflower into florets.
In a saucepan, melt butter over medium heat.
Add onion to saucepan and cook about 5 minutes until golden, stirring occasionally.
Stir in flour and salt; cook 2 minutes longer, stirring frequently.
Gradually stir in milk, chicken broth, and water; add cauliflower and heat to boiling over high heat.
Reduce heat to low, then cover and simmer about 10 minutes until cauliflower is tender.
Meanwhile, preheat oven to 400°F.
Wash green beans and arrange in a single layer on a cookie sheet. Bake for 10 minutes or until beans begin to brown.
In small batches, purée cauliflower in a high-powered blender at low speed until very smooth; remove the center part of the blender cover before puréeing to allow steam to escape.
Return cauliflower mixture to the saucepan; heat over medium heat until hot, stirring occasionally.
Remove saucepan from heat, then stir in mustard and 1 1/2 cups shredded cheese until melted and smooth.
Garnish soup with remaining cheese. Serve with beans and blackberries on the side.