Cut the baked potato in half and scoop the flesh into a large bowl; discard the skin. Using a fork mash the potato, salt and pepper together until smooth. Mix in 3 tablespoons of the egg (reserve the remaining egg for another use). Sprinkle the flour over the potato mixture. Stir until the mixture forms a dough. Knead until smooth, 30 seconds.
On a lightly floured work surface, divide the dough into 4 equal pieces. Roll each piece between your palms and the work surface into a 1/2-inch-diameter rope (about 20 inches long). Cut the dough into 1-inch pieces. Roll each piece of dough over a wooden paddle with ridges or over the tines of a fork to form grooves in the dough.
Bring a large pot of salted water to a boil over high heat. Add the gnocchi and cook until the gnocchi rise to the surface, about 1 minute. Continue cooking until the gnocchi are tender, 1 to 2 minutes longer. Drain and reserve about 1 cup of pasta cooking water.
For the sauce: In a large nonstick skillet, heat the oil over medium-high heat. Add the olives and cook for 1 minute. Add the tomato sauce and bring to a simmer. Add the gnocchi to the skillet and stir until heated through, about 1 minute. Transfer to a large serving bowl. Add the Parmesan and basil. Toss gently until all the ingredients are coated, adding the reserved pasta water as needed to loosen the sauce, and serve.
To bake the potato, using a fork, prick the skin of the potato all over. Wrap the potato in foil and bake at 350 degrees F until tender, about 1 hour. Alternatively, pierce the potato all over with a fork and microwave on full power until tender, turning once, about 12 minutes.