Add first 6 ingredients to a 1-1/2- or 2-pound bread machine according to the manufacturer's directions. Select the dough cycle. When cycle is complete, remove dough from machine. Punch down. Cover and let rest 10 minutes.
Divide dough into 9 portions. Working quickly, shape each into a smooth ball. Punch a hole in center; pull gently to make a 2-inch hole. Place on a greased large baking sheet. Cover; let rise for 20 minutes (start timing after first bagel is shaped). Broil, 5 inches from heat, 3 to 4 minutes, turning once (tops should not brown)
Meanwhile, in large pot bring the 6 cups water and 1 tablespoon sugar to boiling. Reduce heat. Add bagels, 4 or 5 at a time, and simmer for 7 minutes, turning once. Drain on paper towels. (If some bagels fall slightly, they may rise when baked.) Place on a well-greased large baking sheet. Mix the egg white and the 1 tablespoon water; brush over bagels. If desired, sprinkle with poppy or sesame seeds. Bake in a 375 degree F oven for 25 to 30 minutes or until tops are golden. Remove from baking sheets; cool on racks.