Preheat oven to 375 degrees F. Grease or butter a 2.5-quart casserole dish.
Cut peeled potatoes into large 2-inch chunks. Add them to a large pot, cover with cold water, and season with 1 tablespoon (3 teaspoons) kosher salt. Bring to a boil, then reduce heat and simmer for 12 to 15 minutes, until the potatoes are fork-tender.
Drain potatoes in a colander and let them sit for a few minutes to steam off excess moisture.
Use a potato ricer to rice the potatoes into the pot. You also can use a masher, but avoid overworking the potatoes to keep them fluffy.
Warm the milk and butter in a small saucepan until butter is just melted. Pour the mixture into the potatoes. Add sour cream, remaining 1 teaspoon salt and ½ teaspoon black pepper. Fold gently to combine.
Stir in about ¾ cup of the shredded cheese, until fully incorporated.
Transfer the cheesy mashed potatoes to the prepared baking dish and smooth the top.
Sprinkle the remaining cheese evenly over the potatoes.
Bake for 20 to 25 minutes, until the edges are bubbling and the top is golden. For an extra-crispy finish, broil on high for 2 to 3 minutes, watching closely.
Let the casserole rest for 5 to 10 minutes, and serve warm.