Place the eggs, vanilla, vegetable oil, 1 teaspoon of the sugar, and the flour and salt in the bowl of an electric mixer fitted with the paddle and blend on low speed until incorporated. Then beat on high speed for 5 minutes.
Remove the paddle and scrape the batter down the sides of the bowl. Rest the dough in the bowl, covered, until soft and spongy outside, about 1 hour. Then remove it from the bowl, it will be sticky, and make a ball out of it.
Preheat oven to 350 degrees F (175 degrees C) and grease 2 baking sheets.
Sprinkle a work surface with the 1 cup sugar, about 1/8 inch deep.
Place the dough in the center, flatten it slightly with a rolling pin and sprinkle the dough liberally with sugar.
Roll the dough to a thickness of 1/8 inch, a rectangle about 18 by 12 inches. Then, using a pastry cutter or a dull knife, cut out the dough into strips 3/4 inch wide and 2 inches long. Lift each strip, twist in the middle to make a bow tie and place on the cookie sheets, leaving 1/2 inch between each strip.
Bake the cookies for 25 to 30 minutes on the middle rack of the oven, until the cookies are hard to the touch on all corners and golden brown. (If using 1 oven, put the cookie sheets on the top and center racks, then switch them midway). To test for doneness, break a cookie in half. If it is doughy or too soft, it is not done yet. Return to the oven for a few minutes more.
