There's no mistaking the difference between packaged store-bought bagels and ones that are fresh-baked. So although it takes a little time, making a batch from scratch can be quite rewarding and fun. After shaping the dough into balls and poking holes in the centers, your kids will get a kick out of watching their homemade bagels quickly boil and then bake to perfection.
Pour 1/4 cup of the water into a small bowl and sprinkle the yeast over it. Stir with a fork, then let the mixture sit for 5 minutes so that the yeast can thoroughly dissolve.
Combine the remaining 1 1/4 cups of water and the dissolved yeast in a large mixing bowl. Stir in 3 tablespoons of the sugar, then stir in the salt. Add 2 cups of the flour and stir by hand for 100 strokes to blend well.
Stir in enough of the remaining flour, about 1/4 cup at a time, to make a dough that's firm enough to knead.
Turn the dough out onto a floured surface and knead it with floured hands for about 10 minutes, using as much of the remaining flour as necessary to keep the dough from sticking. When fully kneaded, the dough will be firm but supple and should hold the imprint of your hand.
Transfer the dough to an oiled glass or ceramic bowl, turning it a few times to coat the entire surface. Cover the bowl tightly with plastic wrap and place it in a warm, draft-free spot until the dough has doubled in bulk, about 1 hour.
When the dough is ready, punch it down. Turn it out onto a floured surface and knead it for 1 minute. Divide the dough in half, then divide each half into 4 equal pieces.
Shape the 8 pieces into balls, placing them on a floured work counter and dusting the tops with flour. Let them rest for 2 to 3 minutes, then flatten them slightly with your fingers.
Fill a large soup pot about two thirds with water. Add the remaining 1 tablespoon of sugar and begin heating the water to bring it to a boil.
Working with 1 ball of dough at a time, press your floured thumb down through the center to make a hole. Stretch the dough slightly so the hole is about 1 inch in diameter. It will look like a doughnut. Set the bagels aside on a floured surface, covering them loosely with plastic wrap or waxed paper.
Let the bagels rest for 10 minutes while the water comes to a full boil. Meanwhile, grease a large baking sheet and set it aside. Heat the oven to 400°.
Working with the first bagels you shaped, gently drop 2 into the boiling water (a parent's job). They should float, but don't worry if they sink initially - they'll quickly rise to the surface. Boil the bagels for 30 seconds on one side, then carefully turn them with a long-handled, slotted spoon and boil them for another 30 seconds on the other side.
Using your slotted spoon, transfer the bagels to a clean tea towel to drain. Wait a few seconds, then transfer them to the baking sheet, leaving space between them. Repeat with the remaining bagels.
Make the glaze by whisking the egg white and water in a small bowl until frothy. Using a pastry brush, coat the surface of the bagels with the glaze. Sprinkle on sesame seeds or poppy seeds, if desired.
Bake the bagels on the center oven rack until they are deep golden brown, about 22 to 25 minutes. Then transfer the bagels to a wire rack to cool. Makes 8 bagels.