Melt the butter in a large Dutch oven or heavy pot over medium heat. Add the diced onion and cook until softened, about 4 minutes.
Add the garlic and sliced kielbasa; stir and cook for another 2-3 minutes until the sausage begins to brown slightly at the edges and you can smell the smoked paprika release its aroma.
Stir in the flour and cook for 1 minute, whisking constantly so it doesn't clump. The mixture should look like wet sand.
Pour in the broth and beer slowly while whisking to fully incorporate the flour. Bring to a gentle simmer — do not boil hard at this stage.
Stir in the milk, smoked paprika, and pepper. Simmer for 5 minutes, stirring occasionally, until the soup thickens enough to coat the back of a spoon.
Reduce heat to low. Add 2 cups of the shredded cheddar one handful at a time, waiting until each addition melts before adding more. This prevents the cheese from becoming grainy or clumpy.
Taste and season with salt as needed — the kielbasa is already salty, so taste before adding more. Serve topped with the remaining 2 cups of cheddar.
