Put the potatoes in a saucepan; cover with cold water and add 2 tablespoons kosher salt. Bring to a simmer over medium-low to medium heat and cook, uncovered, until tender, about 45 minutes. Drain the potatoes and let sit until just cool enough to handle, then peel and transfer to a bowl.
Add the butter to the potatoes. Add the yogurt, 2 teaspoons kosher salt, and pepper to taste; mash with a potato masher or fork.
Stir the chipotle peppers. Fold in the goat cheese. Spoon the mashed potatoes into a serving dish and sprinkle with the caramelized shallots.