Use a 6-quart slow cooker. Spread four 2-foot lengths of aluminum foil or parchment paper onto your kitchen countertop. Evenly divide the sliced fingerling potatoes into the center of each piece. Place a salmon fillet onto each pile of potatoes. Sprinkle dill, garlic, salt, and pepper onto the top of each piece of fish.

Divide the green beans into 4 portions and put them on top of the salmon. Top each pile of green beans with 2 tablespoons of butter. Fold over the edges of the foil or parchment paper and crimp the sides to create a fully enclosed packet.

Place the packets into an empty 6-quart slow cooker (do not add water). Cover and cook on high for 2 hours. Check doneness by opening one packet (careful, steam will escape quickly!). The fish is finished cooking when it flakes easily with a fork.