Chop the potatoes into 1-inch chunks, slice the carrots and celery thinly, and mince the garlic. Prepare all your ingredients for easy assembly.
Heat a skillet over medium heat and add the sausage. Cook until it’s nicely browned and caramelized, breaking it apart with a spoon as it cooks for about 8 minutes. Drain excess fat if needed and transfer the sausage to the slow cooker.
Add the chopped potatoes, carrots, and celery into the slow cooker on top of the sausage. Pour in the broth and stir in the minced garlic, thyme, and bay leaf. Season with salt and pepper.
Cover the slow cooker and set it to low. Let it cook for 6 to 8 hours until the potatoes are tender and the flavors meld into a fragrant, hearty broth.
Check the soup after about 6 hours: the potatoes should be easily pierced with a fork, and the broth will smell savory and rich. Remove the bay leaf and discard.
Taste the soup and adjust the seasoning with more salt and pepper as needed. For a bright finish, stir in chopped parsley.
Ladle the steaming, thickened soup into bowls and enjoy immediately, savoring the smoky, savory flavors and tender potatoes.