water in a large bowl; let stand 5 minutes. Stir in sugar, salt, and 5 1/2 cups flour until moistened.
Spread 1/2 cup flour on a board and scrape dough onto it. Knead dough until very smooth and elastic, 10 to 15 minutes; add more flour as required to prevent sticking.
Place dough in a lightly oiled bowl, cover with plastic wrap, and let rise in a warm place until almost doubled, about 40 minutes.
When dough is almost risen, bring 3 quarts water to boiling in a covered 4- to 5-quart pan over high heat.
Meanwhile, knead dough on a lightly floured board to expel air, then divide into 12 equal pieces. To shape bagels, knead each piece into a smooth ball. Holding the ball with both hands, poke your thumbs through the center to form a hole. With 1 thumb in the hole, work around the bagel to make it smooth and evenly thick.
Set bagels on lightly floured 12- by 15-inch baking sheets; let stand, covered with plastic wrap, just until they are slightly puffy, about 10 minutes.
With your hand, gently slip bagels into boiling water, adding 4 at a time to pan. Adjust heat to maintain a gentle boil and turn bagels often until they feel firm but are still slightly spongy, 3 to 5 minutes. Chill remaining bagels until you are ready to boil them.
With a slotted spoon, transfer bagels to a towel; drain briefly, then place bagels slightly apart on 2 lightly oiled 12- by 15-inch baking sheets.
In a food processor or with a knife, coarsely chop cheese.
Beat egg yolk to blend with 1 tablespoon water and brush mixture over bagels. Then pat cheese onto bagels, pressing gently to make the cheese stick.
Bake bagels in a 375 oven until browned, 30 to 35 minutes. Cool on racks.
