PREPARATION:

1. Combine 1/4 cup of the warm water, yeast and 1 teaspoon sugar. Stir to dissolve yeast and let stand until bubbly, about 5 minutes.
2. Fit processor with steel blade. Measure flour, oil and salt into work bowl. Cover; process until mixed, about 5 seconds. Add yeast mixture and 1 egg; process until blended, about 10 seconds.
3. Turn on processor and very slowly drizzle just enough remaining warm water through feed tube so dough forms a ball that cleans the sides of the bowl. Process until ball turns around bowl about 25 times. Turn off processor and let dough stand 1 to 2 minutes.
4. Turn on processor and gradually drizzle in enough remaining warm water to make dough soft, smooth and satiny but not sticky. Process until dough turns around bowl about 15 times.
5. Turn dough onto lightly greased surface. Shape into ball and cover with plastic wrap. Let stand about 15 minutes.
6. Divide dough into 12 equal pieces. Shape each piece into a strand about 6 inches long. Bring both ends of each strand together to form a doughnut shape. Moisten ends and pinch together to seal. Place bagels on greased cookie sheet and let stand at room temperature about 15 minutes.
7. Combine remaining 2 quarts water and remaining 2 tablespoons sugar in Dutch oven or stock pot. Bring water to a boil. Gently place 3 or 4 bagels at a time in boiling water. When they rise to the surface, turn them over and cook until puffy, 1-1/2 to 2 minutes longer. Remove bagels from water with a slotted spoon and place on greased cookie sheet.
8. Heat oven to 425°F. Beat remaining egg and 2 tablespoons cold water with a fork. Brush mixture over bagels. Bake until crusts are golden and crisp, 20 to 25 minutes. Remove from cookie sheet. Cool on wire rack.