Put potatoes in a large pot and add enough cold water to barely cover. Stir in 1 tsp. salt. Bring to a boil over high heat, covered. Reduce heat to medium and boil gently until potatoes are tender when pierced, about 15 minutes. Drain; return to pot, off the heat.
In a medium bowl, combine yogurt, 1/2 tsp. salt, and 1/3 cup each parsley, dill, and chives. Set aside.
Mash hot potatoes with a potato masher until all large chunks are broken and potatoes are fluffy. Add milk, 2 tbsp. butter, 1/2 tsp. salt, and the pepper; mash to blend. Stir herbed yogurt into potatoes. Add salt and pepper to taste. Transfer to a serving bowl, top with remaining 1 tbsp. butter, and scatter 2 tbsp. mixed chopped herbs over the top.
Make ahead: Through mashing of potatoes in step 3 (before adding yogurt), up to 1 day, chilled. Reheat in a microwave until hot (if they've stiffened up, add enough milk to make them creamy). Stir herbed yogurt into warmed potatoes and proceed as directed.