Place potatoes in a large pan, cover with water and stir in 1 tsp. salt. Leave uncovered and bring to a boil over high heat. Reduce heat to medium-low and simmer until very tender, about 15 minutes. Drain well.
Transfer potatoes to a bowl; mash until smooth. Stir in Parmesan and butter, then chives and sour cream. Season with salt and pepper. When cool enough to handle, divide mixture into 8 portions and form into patties, each about 1 inch thick. Place on a baking sheet. (Can be made up to 24 hours ahead. Cover and refrigerate.)
Preheat oven to 375°F. Warm 2 Tbsp. oil in a large nonstick skillet over high heat until hot but not smoking. Add 4 patties; cook until brown, about 2 minutes per side; transfer carefully to baking sheet. Wipe out skillet. Repeat with remaining 2 Tbsp. oil and remaining patties. Bake patties until heated through, about 5 minutes.