Melt butter in a saucepan over medium-high heat. Add onion and cook, stirring often, until soft and translucent, about 5 minutes. Stir in flour and cook for 1 minute, stirring constantly. Whisk in milk and mustard, bringing to a boil. Reduce heat to simmer and cook until thick and creamy, 6 to 8 minutes.

Stir in Cheddar and Velveeta, cooking until melted. Add chicken stock, cream, carrots, and broccoli. Cook until vegetables are tender, 10 to 15 minutes, stirring occasionally. Season with salt, pepper, and hot sauce and cook for 2 more minutes. Remove from heat and serve with crusty bread.