Mound the flour on a clean work surface. Sprinkle with the salt and toss with your fingers, then make a well in the center of the flour and pour in the beaten egg. Quickly combine with your fingers and as the dough starts to come together, add 1 tablespoon of water at a time, up to 3 tablespoons total. Keep kneading until you get a solid but soft ball of dough, about 5 minutes. Place it in a bowl and cover with a damp towel for 30 minutes to 1 hour.
Divide the dough into 4 equal pieces. Roll them each out on a floured surface to make a skinny log about 10-inches-long. Pinch off a small piece of dough, just enough to cover your fingertip, and roll the dough against the floured work surface with your thumb, creating a "little ear". Transfer the orecchiette to a large piece of parchment paper. Keep pinching and making the orecchiette with your thumbs until no dough is left.
In the meantime, bring a pot of salted water to a boil. Pick up the sides of the parchment paper to slide the fresh pasta into the water. Boil until the pasta floats to the top, 2 to 3 minutes.
Reserve about half a cup of pasta water, and drain the rest.
Combine the pasta with the oil and 2 tablespoons pasta water in a large bowl (add more pasta water if it seems dry!). Season with a pinch of salt and pepper and divide between 2 dinner bowls, or among 4 smaller bowls. (Our kids eat only a tiny bit, so we could still call this 2 servings.)
Garnish with freshly grated Parmesan and fresh basil leaves!