1. Sift flour into mixer bowl or another large bowl. Make a well in center of flour. Pour in ½ cup lukewarm water. Sprinkle yeast on top and add 1 teaspoon sugar. Leave 10 minutes until yeast is foamy. Add another 2 teaspoons sugar, oil, 3 of the eggs, remaining ½ cup plus 2 tablespoons water, and salt to well in center of flour. Mix with dough hook or with a wooden spoon until ingredients begin to come together into a dough. If using mixer, beat dough until smooth. If mixing with a spoon, when adding flour becomes difficult, mix in remaining flour with your hands.
2. Knead dough vigorously on a work surface about 10 minutes or until very smooth and no longer sticky. Put dough in a clean oiled bowl, cover with a damp cloth, and let rise in a warm place about 1 hour or until light but not doubled.
3. Shape bagels (see Shaping Bagels. below). Then, preheat oven to 400°F. To boil bagels, bring 2 quarts water and 1½ tablespoons sugar to a boil in a large saucepan. Add 3 or 4 bagels and boil 1 minute. Turn them over and boil 1 minute. If holes begin to close, force them open with handle of a wooden spoon. With a slotted spoon, transfer them to a cloth or to paper towels. Repeat with remaining bagels.
4. Put bagels on 2 or 3 lightly floured or greased baking sheets. Beat remaining egg with a pinch of salt. Brush egg over bagels. Sprinkle with seeds, if using.
5. Bake bagels about 20 minutes or until browned; if baking sheets don’t fit on center rack, bake them one above the other and switch their positions after 10 minutes. If not serving them right away, cool them on a rack and wrap them
To Shape Bagels:
1. Make bagel dough and let it rise. (If you prefer, you can make the dough 1 day ahead, let it rise 30 minutes and punch it down. Then cover it with a damp cloth or plastic wrap so it doesn’t dry out and refrigerate it overnight. Let it come to room temperature before shaping bagels.)
2. Knead the risen dough lightly. Roll dough into a thick log. For every cup of flour, cut into 3 pieces with a floured knife to make 3 bagels. (4 cups of flour yields 12 pieces.)
3. Roll each piece of dough into a very smooth ball by holding it under your cupped palm on an unfloured surface, and rolling it over and over while pressing it firmly on the surface. The more the dough is rolled, the more evenly shaped and smooth the final bagel will be.
4. Flatten one ball slightly. Make a hole by flouring your index finger and pushing it through the center of the dough ball. Twirl the dough around your finger on the work surface to stretch the hole; then insert 2 fingers and continue twirling until hole is about 1 inch in diameter and evenly round.
5. Gently pull the edges to even out the shape of the bagel. Transfer the bagels to a floured board. Cover and let rise 15 minutes. Boil and bake them as directed in each recipe.