In a medium bowl whisk together the sugar, egg whites, and almond or vanilla extract until combined. Whisk in the oil and then the flour, ginger, and salt until well combined and free of lumps. Pour the batter into an airtight container and refrigerate for at least 4 hours or up to 1 week. The batter needs to sit for a bit to fully absorb the flour into the whites.
Heat the oven to 325°F and place a rack in the center of the oven. Line a very flat baking sheet with a silicone baking mat or parchment paper liberally sprayed with non-stick spray . (It will be infinitely easier with a silicone baking mat but parchment does work.) Write out 16 tiny fortunes on little slips of paper about [1/4]-inches by 2-inches. Fold the fortunes into an teeny tiny square. Set aside.
Spoon 1 rounded tablespoon of batter onto the baking sheet and use the back of a small butter knife or an offset spatula to spread the batter evenly and as thinly you can into a circle about 5 inches in diameter. It takes some practice to get the batter nice and thin. Do your best and if you need to use your fingers to smear the batter a bit before spreading with the knife or offset spatula. Bake until the cookie is golden brown all over, 12 to 16 minutes.
Okay here is the fun part! Read these directions a few times before trying to make sure you understand how to shape the cookies. You have about 10 seconds after the cookie has come out of the oven during which it is still pliable. This 10-second window coincides with when the cookie is at its hottest and is most difficult to handle. Try wearing latex or plastic gloves to shield your fingers from the heat. Moving quickly, remove the baking sheet from the oven and use an offset spatula to quickly remove the cookie from the baking sheet. The easiest way to do this is to swipe the offset under the cookie very fast to loosen the whole thing. Time matters! Place one of the fortunes in the middle of the cookie, fold the cookie in half, hold the half-moon-shaped cookie in both hands with the rounded half facing up to the sky and push the base of the half-moon with both thumbs upwards to make the cookie bend in half into the characteristic fortune cookie shape. Hold the cookie until it cools and hardens into shape or put it pointed side down into an empty egg carton which is the perfect shape for holding the cookies. Whew! Take a deep breath and congratulate yourself. Repeat until all of the batter is used up. Once you are more adept with the technique you can make 3 to 4 at a time. Cookies may be made up to 3 days in advance and stored in an airtight container at room temperature.