Melt the butter in a large pot over medium heat. Add the onions and saute until translucent or just starting to brown. Add the garlic and celery and saute until fragrant, about 2 more minutes. Stir in the mustard and toss so that everything is well coated.
Add the chopped potatoes to the pot and cover in the chicken broth. Cover and bring to a simmer. Let simmer for about 20 minutes, or until the potatoes are easily pierced with a fork. Add the ham and simmer for an additional 5 minutes.
Turn the heat off, add the lemon juice, salt, and pepper. Stir to combine and taste to determine if it needs any additional seasoning.
Garnish with the chives and enjoy!