Prep the base: Add broccoli, onion, garlic, smoked paprika, dried mustard (if using), and butter to the crockpot. Season with a generous pinch of salt and pepper.
Add the liquid: Pour in the broth.
Stir lightly to distribute the seasonings and butter.
Slow cook: Cover and cook on Low for 5–6 hours or on High for 2½–3 hours, until the broccoli is very tender.
Blend to your liking: Use an immersion blender to blend directly in the crockpot until the soup is mostly smooth, leaving some small pieces for texture. If you prefer it chunkier, pulse a few times. If you don’t have an immersion blender, carefully transfer half to a blender, blend, and return it to the pot.
Enrich the soup: Stir in the cream cheese until melted and smooth.
Add heavy cream and stir again.
Add the cheese: Reduce heat to Warm or Low. Gradually add the shredded cheddar by the handful, stirring after each addition until melted and silky. Avoid boiling—high heat can cause cheese to separate.
Adjust seasoning: Taste and add more salt, pepper, or a pinch of red pepper flakes for heat.
If you want extra richness, swirl in a bit of sour cream.
Load it up: Ladle into bowls and top with crispy bacon, green onions or chives, a sprinkle of extra cheddar, and a dollop of sour cream if you like.