In a large bowl, combine the water, 3 tablespoons of the sugar, and yeast. Stir and let stand until foamy, about 5 minutes.
Gradually add 4 cups of the flour and the salt to the mixture and stir well with a wooden spoon just until the mixture comes together.
Add 1 to 1 1/2 cups of the remaining flour, 1/2 cup at a time, until a stiff dough is formed. If dough is too stiff to mix with a spoon, transfer it to a standing electric mixer fitted with a dough hook and knead the dough.
Turn dough out onto a lightly floured surface and knead with your hands until smooth and no longer sticky, about 5 minutes. If dough seems sticky, add a bit of the remaining flour until it is no longer so. This should be a very stiff and heavy dough.
Lightly oil a large bowl with 1 teaspoon of the vegetable oil and place the dough in the bowl, turning to coat. Cover the bowl with a damp clean kitchen towel or plastic wrap and set aside to rise in a warm, draft-free area until dough has almost doubled in size, about 1 hour.
Remove the dough from the bowl and punch dough down with your fist.
Divide the dough into 12 equal pieces and form each piece of dough into a ball. Stick your finger through the middle of the dough to make a hole. Using your fingers, smooth the top and sides of the dough and pull the sides gently apart to enlarge the hole and make a circle about 3 to 3 1/2 inches in diameter. Repeat with the remaining pieces of dough.
Place the bagels on a lightly floured surface, cover with a damp clean kitchen towel and set aside to rest in a warm, draft-free area until dough has risen but not doubled, about 15 to 20 minutes.
Position rack in center of oven and preheat the oven to 400ºF. Lightly grease a baking sheet with the remaining teaspoon of vegetable oil and sprinkle lightly with the cornmeal.
In a large, heavy pot, bring 12 cups of water to a boil with the remaining tablespoon of sugar.
Carefully lower the bagels into the boiling water in batches, leaving enough room for them to expand as they cook (usually 3 or 4 per batch, depending on the size of your pot). Cook for 5 minutes, turning once midway through the cooking time using a slotted spoon.
Using a slotted spoon or skimmer, remove the boiled bagels and set aside to drain on paper towels.
Transfer bagels to the prepared baking sheet with the slotted spoon and brush each with the beaten egg mixture. Sprinkle with the optional toppings, if desired, and bake until golden brown and crusty, about 25 to 30 minutes.
With oven mitts or pot holders, remove the baking sheet from the oven and transfer the bagels to a wire rack to cool.
YIELD: 12 bagels