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.