In the slow cooker pot, add the potatoes, onion, celery, carrots, garlic, salt, pepper and chicken broth. Stir together.
Slice (widthwise) half of the quantity of sausage. Keep the other half of the sausage in a whole link so that it's easier to remove from the pot before mashing the cooked vegetables.
To the pot, add the halved link of sausage and top the ingredients that are in the pot with the sausage. Don't stir in the sausage. Keep the sausage on top so that it's easier to remove before mashing the veggies. Top with a couple pinches of pepper. Cover with a lid, set the slow cooker on high, and cook for 4 hours.
After it's done cooking, turn the slow cooker setting to warm or low and transfer the sausage to a plate. Set aside for a moment. Using a potato masher, mash the potatoes and vegetables until the consistency is thick and still has small pieces in it. Slice the remaining link of sausage widthwise and add to the pot. Stir together.