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Cook potatoes in a large pot of boiling water over medium-high heat until tender when pierced with the tip of a knife, 25-30 minutes; drain. As soon as potatoes are cool enough to handle, peel and pass through a potato ricer into a large bowl. Let cool.
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Sprinkle 1¼ cups flour and 2 teaspoons salt over potatoes and, using your hands, make a well in the center. Pour egg into the well and stir in with a wooden spoon. Turn out dough onto a floured surface and gently knead, dusting with more flour as needed, until smooth but not elastic (be careful not to overwork), about 2 minutes.
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Divide dough into 8 pieces. Roll each piece into a 12"-long rope about 1" thick. Cut into 1" pieces, dust with flour, and arrange in a single layer on a lightly floured rimmed baking sheet.
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Working in batches, cook gnocchi in a large pot of boiling salted water until they float to the surface, about 2 minutes. Using a slotted spoon, remove from pot and transfer to a bowl.
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Add half of butter to a skillet over medium-low heat. When butter has melted, add shallot and cook for 1 minute. Add wine and bring to a simmer. Cook until wine has reduced by half, then add lemon juice. Add remaining two tablespoons of butter, 1/2 cup of reserved cooking liquid, gnocchi and peas, tossing gently to coat gnocchi in sauce. Cooking just until peas are warmed through, about 1-3 minutes. Remove from heat and add chives, parsley and Pecorino Romano. Season with salt and pepper and toss gently once more. Divide into bowls and serve with more Pecorino.