Wash and cut the potatoes into halves or quarters depending on their size. Place potatoes in a pot and cover with cold water by at least 1 inch above the potatoes and add 1 tbsp kosher salt. Bring to a boil on the stove and boil until they are very fork-tender. Drain and allow potatoes to dry and cool until you can touch them.
In a large bowl, combine yogurt, capers, shallot, garlic, lemon juice, olive oil, dill, and chives. Mix to combine. Taste and add salt and pepper to your liking.
Once potatoes are cool enough to touch, but still warm, add them to the dressing, then fold in until the potatoes are fully coated. Place in the fridge and let cool for a few hours, or until the potatoes have absorbed the dressing and it is no longer soupy.
Serve chilled and garnish with any remaining chives or dill.