How to Make It
In a medium saucepan, cover the potatoes and bring to a boil. Simmer over moderately high heat until the potatoes are tender, about 8 minutes. Drain the potatoes, then return them to the pan; shake over moderately high heat until dry.
Working over a large rimmed baking sheet, rice the hot potatoes in an even layer. In a small bowl, whisk the egg yolks with the lemon zest, 1 teaspoon of olive oil and the salt and pour over the potatoes. Sprinkle the flour over the potatoes and stir gently just until a dough forms.
Gently roll the dough into four 1/2-inch-thick ropes. Using a sharp knife, cut each rope into 1/2-inch pieces. Roll each piece against the tines of a fork to make ridges. Transfer the gnocchi to the baking sheet, cover with plastic wrap and refrigerate.
In a small saucepan, bring the chicken broth to a simmer. Remove from the heat and whisk in the 1 stick of butter, a few pieces at a time, until the sauce is creamy. Warm the sauce on low heat if necessary. Stir in the lemon juice and season with salt.
In a large pot of boiling salted water, cook the gnocchi until they rise to the surface, then cook them for 1 minute longer. Gently drain the gnocchi, toss with the remaining 2 tablespoons of olive oil and transfer to a baking sheet until cool.
In a large nonstick skillet, melt 1 tablespoon of the butter. Add half of the gnocchi and cook in a single layer over high heat until browned on the bottom, 2 minutes. Transfer to a large bowl and repeat with the remaining 1 tablespoon of butter and gnocchi.
Reheat the sauce; pour it over the gnocchi and fold gently with a rubber spatula until they are evenly coated. Transfer to a platter and garnish with the chives. Grate the bottarga on top, if desired. Serve.