In small bowl, stir yeast and the 1 tablespoon sugar into the 1/4 cup warm water. In a food processor bowl (must be a 5-cup capacity bowl or larger), combine flour, the 1/4 cup sugar, and the salt. Cover and process for 30 seconds. With food processor running, slowly add yeast mixture and egg-oil mixture through feed tube. Continue to process until combined. Slowly add enough of the 1/3 cup water to the mixture to make a dough that almost forms a ball.
Punch down dough. Turn dough out onto a lightly floured surface. Divide into three equal portions. Cover and let rest for 10 minutes. Lightly grease a large cookie sheet; set aside.
Roll each portion of dough into a 15-inch rope. Place the ropes, 1 inch apart, on the prepared cookie sheet. Loosely braid the ropes, working from center to each end. Press the ends together to seal; tuck ends under loaf. Cover; let rise in a warm place until nearly double (about 30 minutes). Combine the slightly beaten egg and the 1 teaspoon water. Brush loaf with egg mixture.
Bake in a 350 degree F oven about 25 minutes or until bread sounds hollow when you tap the top of the loaf with your fingers. Immediately remove bread from cookie sheet. Cool on wire rack. Makes 1 braid (24 servings).