Step 1In a stand mixer fitted with the dough hook attachment, combine 3 cups water with the sugar, vegetable oil and 3 cups of the flour. Mix it at low speed until combined.
Step 2Add the yeast and another 3 cups flour and continue to knead the dough at low speed until all of the flour has been incorporated.
Step 3Add the salt and the remaining flour. Knead the dough at medium speed until it's soft and smooth and it comes away from the sides of the bowl. If the dough is sticky add 1 tablespoon of flour at a time, until smooth and no longer sticky.
Step 4Roll the dough into a ball. Place the dough in a lightly greased bowl, cover with a lightweight cloth or loosely with plastic wrap. Leave the dough for about 30 minutes to 1 hour, depending on room temperature, until it doubles in volume.
Step 5Punch down the dough. On a lightly floured surface, divide the dough into 12 even balls. (If dough is reverting when shaping, let it rest for a minute and start the process again.) To keep the dough from drying out, place a damp towel on top.
Step 6Roll each ball to about 15 inches in length. Press and roll the ends together to form a bagel.
Step 7Heat the oven to 425 degrees. Place 4 to 6 bagels on a parchment-lined cookie sheet, leaving 1 to 1 1/2 inches between each bagel as they will spread. Cover the bagels with a cloth. Allow them to grow by half again in size, about 15 minutes. They should be light and fluffy. Widen the holes in the bagels to 1 1/2 inches each, as they will shrink while baking.
Step 8In a small bowl, beat the eggs, then brush bagels with the egg wash. Bake until golden, about 20 to 22 minutes.