Make the gnocchi: Mix the instant potatoes with 1 cup hot water in a bowl using a fork; set aside until the potatoes absorb the liquid, 3 minutes. Meanwhile, puree the basil and 1/4 cup cold water in a blender until smooth. Stir the basil puree, egg, pecorino and 1 tablespoon salt into the potato mixture.
Mound 3/4 cup flour on a surface; add the potato mixture and knead together until smooth, gradually adding the remaining 1/4 cup flour. Divide the dough into 3 portions; roll each into a 1-inch-diameter log (dust with flour, if needed). Cut each log into 1/2-inch pieces; place on a parchment-lined baking sheet and refrigerate while making the sauce (or freeze up to 1 month).
Make the sauce: Heat the olive oil in a large skillet over medium-high heat. Add the pine nuts; fry until toasted, swirling the pan. Transfer the nuts with a slotted spoon to paper towels and season with salt. Add the shallots and garlic to the pan and cook until soft, 3 minutes. Add the wine and bring to a boil. Add the tomatoes, season with salt and pepper and toss; cook until the tomatoes are just soft, 2 more minutes.
Cook the gnocchi in a large pot of salted boiling water until tender, about 1 minute after they float to the surface. Drain and toss with the sauce. Top with the toasted pine nuts and more basil and pecorino.
Photograph by Tina Rupp
Recipe courtesy Ashley Archer for Food Network Magazine