Melt the butter over medium-high heat. Add onions, garlic, carrots and broccoli. Sauté for 5 mins until vegetables are softened, stirring often as to not burn the garlic.
Sprinkle in the flour and stir for 1-2 mins until the flour no longer smells raw.
Slowly add in broth, milk, and cream, while stirring constantly to make sure there are no clumps.
Add salt, pepper, paprika, red pepper and nutmeg. Bring to a simmer and cook for 10-15 mins until thickened and vegetables are cooked through.
Turn off the heat, stir in cheese until melted and smooth.
Serve with warm french bread!