In the bowl of a stand mixer fitted with the dough hook attachment combine the oil, wine, salt and yeast. Mix on low speed for 1 minute. Add in the flour and continue mixing on low speed until a smooth ball of dough forms, about 10 minutes. If the mixture seems dry add 1 tablespoon of wine at a time until it comes together.
Attach the pasta roller attachment to the mixer and set it to the largest setting possible. With the mixer running on the highest speed, take a tennis ball sized amount of dough and flatten it with your hands as much as possible. Feed it through the pasta roller holding the dough taunt. The dough will have holes for the first passing. Pass the dough through the machine 20 more times. The dough is ready once it feels smooth, supple and almost velvety in your hands.
Take the piece of dough to a clean work surface and roll it into a rope about 1/2 inch thick. Cut pieces from the rope about 8 inches long, the length of a dinner knife is the perfect measurement! Join the ends of the taralli to form a round shape. Place one end on top of the other and push down hard with your index finger to seal them together. Set the taralli aside on a clean kitchen towel or tablecloth not touching and repeat passing the dough through the machine and rolling it out until you have made all your taralli.
* You will be able to make 2-3 taralli at a time. You must work in small batches passing the dough through the machine because it must be rolled out immediately or it will dry out.
Once you have made all your taralli let them rest for 15 minutes before boiling them.
Put a 5 quart pot over a high flame and bring 3 quarts of water to a boil. Drop the taralli in 4-5 at a time, beginning with the first ones made. Stir the pot with a wooden spoon to prevent sticking. As soon as the taralli rise to the surface scoop them out and lay them back on the towel.
Preheat the oven to 475 degrees.
Hold the boiled taralli in your hand and with a sharp knife make an incision 3/4 of the way through.
Place the taralli directly on the rack of the oven. Bake for 7 minutes at 475 and then lower the temperature to 400 degrees. Bake for an additional 10 minutes turning the taralli with tongs halfway through.
Once fully baked cool the oven down to 180 degrees. Place the taralli on a 9x13 inch baking sheet and bake for 30 minutes. This will remove all the humidity from the taralli and help them stay fresh much longer.