PREPARATION:
- 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.
- 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.
- 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.
- 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.
- Turn dough onto lightly greased surface. Shape into ball and cover with plastic wrap. Let stand about 15 minutes.
- 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.
- 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.
- 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.