Mix flours and salt in a large, round bottomed bowl big enough to knead the dough.
Form a well in the center of the flour and add the egg and oil.
Use a spoon to mix the egg and oil into the flour. The mixture will be very crumbly.
Add the water one tablespoon at a time, mixing between each addition with your hand. Keep adding spoonfuls of water until the dough forms a rough ball.
Knead the ball for a couple minutes. If the ball doesn't want to stay together, add another spoonful of water and knead together thoroughly.
Wrap the ball of dough tightly in plastic wrap and set aside for 30 minutes. This will allow the flours to hydrate.
Pinch off a small bit of dough and squeeze in your hand to make a rough ball. Put the ball in a 1/2 teaspoon measuring spoon and use your thumb to press the dough into a little cup. Pull your thumb and the pasta will come off the spoon easily. (See the picture.) If the big blob of dough starts to dry out, mist it with water. You will make about 250 orechietti.
Set rim-side-up on parchment paper to dry for about two hours. Turn over and let dry another two hours. If not using the same day, store loosely packed in a rigid container (not a bag) in the freezer.
To cook, bring a large pot of salted water to a boil. Add orecchietti and cook for 2 minutes. Test and continue boiling until a pleasant texture is reached. Serve hot tossed in olive oil or sauce.