Whisk together cold water, egg, and salt in a large bowl until salt has dissolved. Mix in dry potato flakes until incorporated. Stir in 1 tablespoon chives; season with black pepper and cayenne pepper.
Divide potato mixture into 4 equal portions and shape into pancakes.
Heat oil and butter in a large skillet over medium-high heat. Swirl the pan to combine.
Place pancakes in the pan, reduce heat to medium, and cook until the bottoms are browned, about 10 minutes. Flip and flatten slightly. Cook until other sides are browned, 7 to 8 more minutes.
Transfer pancakes to a serving plate. Top each with a dollop of sour cream and sprinkle with remaining chives.