To prepare the mousseline: Put the potatoes in a saucepan, cover with cold water, and add the Kosher salt. Bring to a boil over high heat and then turn heat down to medium-low and simmer. Cook the potatoes for about 20 minutes, until soft when pierced with a fork. Drain the potatoes into a fine mesh sieve and then press them through the sieve back into the saucepan. (You can use a food mill or ricer, but the texture will not be as fine). Stir in the butter and milk and reheat the potato mixture over low heat. Add the chives, salt, and pepper. Keep warm until ready to serve.
To prepare the scallops: Pat the scallops dry, cut in half horizontally (into 2 medallions), and season with salt and white pepper. Pour the oil into a heavy skillet and set over high heat until the oil is shimmering. Carefully add the scallops to the hot skillet and sauté for 2 minutes on one side until golden brown and just cooked through. Remove scallops and transfer to a warm plate. Turn down the heat to low and add the lemon juice to the skillet. Stirring with a wooden spoon, deglaze the skillet so that any solids have dissolved. Using a fine mesh strainer, strain the liquid from the skillet into a gravy boat or serving cup.
To serve: Place about 1 teaspoon of the potato puree in the center of each small plate or scallop shell. Place a scallop on top (browned-side up) and evenly distribute the caviar on top of the scallops. Drizzle the lemon sauce from the skillet around the potato puree on each plate.