Prep the potatoes: Scrub the baby potatoes clean and halve any larger ones so all pieces cook at the same rate.
Trim the green beans: Snap or cut off the stem ends of the green beans and set them aside; they go in later.
Slice the sausage: Cut the beef sausage into 1/2-inch thick rounds so each piece browns slightly at the edges as it cooks.
Layer the crockpot: Add the potatoes and sliced onion to the bottom of the crockpot, then scatter the sausage rounds on top.
Add broth and seasoning: Pour in the chicken broth, then sprinkle in the garlic, smoked paprika, salt, and pepper, stirring lightly to combine.
Cover and cook on low: Cover the crockpot and cook on low for 6 hours, until the potatoes pierce easily with a fork and the broth smells deeply savory.
Stir in the green beans: Add the trimmed green beans to the pot during the last 30 minutes of cooking, submerging them in the broth.
Finish with butter: Stir in the butter and let it melt into the broth, coating the vegetables and sausage with a light, silky sheen.
Check doneness: The dish is ready when the green beans turn bright green and tender-crisp and the broth has reduced slightly and thickened.
Serve warm: Ladle into bowls, spooning extra broth over the top so the potatoes and sausage soak up the savory liquid.
