Place thawed hashbrowns, ham, onion, celery, broth, salt and pepper in a 6-quart slow cooker and stir to distribute the potato shreds.
Cover and cook on low for 5 hours until the potatoes and celery are soft. Stir thoroughly from the bottom.
Turn the cooker to high and add cream cheese in small cubes. Cover for 15 minutes, then whisk gently until the cream cheese melts into the broth.
Heat covered for 10 to 15 minutes more until the soup is steaming and ham reaches 165°F.
Switch off the heat and stir in Swiss and Dijon until smooth. Divide among 6 bowls, keeping small ham pieces visible through the creamy potato base.