Heat the oven to 425°F. Place 4 large russet potatoes on a rimmed baking sheet. Rub all over with 2 teaspoons olive oil and season with 1 teaspoon of the kosher salt. Prick each potato several times with a fork.

Bake, flipping halfway through, until soft when pierced with a knife and the skins are dry, 55 to 65 minutes total.

Let the potatoes sit until cool enough to handle, about 15 minutes. Cut off the top ¼-inch of the potatoes. Scrape out the potato flesh from the tops with a spoon into a medium bowl and discard the top skin. Scoop out the flesh from the bottom portion of the potatoes into the same bowl, leaving a ¼-inch-thick shell around the bottom and sides. Return the potato shells to the baking sheet skin-side down.

Pass the potato flesh through a ricer or food mill into a medium saucepan. (Alternatively, mash very well with a fork or potato masher, then transfer to the saucepan.)

Place the pan over medium-low heat. Stir in ½ cup of the heavy cream, 4 tablespoons room-temperature unsalted butter, the remaining ½ teaspoon kosher salt, and ¼ teaspoon black pepper. Cook, stirring constantly, until the butter is melted and the mixture is smooth.

Remove the saucepan from the heat. Stir in 2 tablespoons finely chopped chives. Taste and season with kosher salt as needed.

Stir the remaining 2 tablespoons heavy cream, 1 ounce shredded Gruyère cheese, and 1 large egg yolk together in a small bowl until combined.

Spoon the potato purée evenly into the potato shells. Use a spoon to make a well that’s about ½-inch deep and ¼-inch wide lengthwise through the purée in each potato, pushing the excess purée to the sides. Spoon the egg mixture evenly into the wells.

Bake until the egg and cheese mixture is mostly set, no longer jiggly, and slightly browned, 12 to 15 minutes. Turn the oven to broil and broil until the tops are golden-brown, 3 to 4 minutes. Let cool for 5 minutes and garnish with chopped fresh chives if desired before serving.