For the Potato-Chive Mousseline:
1 pound Yukon Gold potatoes, peeled and cut in half
2 teaspoons Kosher salt
¼ cup softened butter
¼ cup milk
2 tablespoons chopped fresh chives
Salt and freshly ground white pepper to taste
For the Scallops:
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. Turn heat to medium low and simmer until soft when pierced with a fork, about 20 minutes. Drain the potatoes, using a fine-mesh sieve, and then press them through the sieve 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 over low heat, stirring occasionally until ready to serve.
To prepare the scallops: Pat the scallops dry and season with salt and 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 each side until golden brown and just cooked through (do not flip). Remove the scallops and transfer to a warm plate. Turn the heat to low, add the lemon juice to the skillet, and add the butter. Stirring with a wooden spoon, deglaze the skillet to dissolve any solids. Pour the liquid through a fine-mesh strainer into a gravy boat or serving cup.
To serve: Place 2 teaspoons of the potato puree in the center of each small plate (or use scallop shells if available). Place each scallop, browned side up, in the center of the puree and evenly distribute the caviar on top. Drizzle the lemon sauce around the potato puree.