Heat 1/2 cup beer and 1 cup water over low heat in a small, heavy saucepan to between 100° and 110°. Combine beer mixture and yeast in a large bowl, stirring until yeast dissolves. Let stand 5 minutes. Stir in egg white.
Lightly spoon flour into dry measuring cups; level with a knife. Add 4 1/4 cups flour and salt to beer mixture; stir until a soft dough forms. Turn dough out onto a floured surface. Knead until smooth and elastic (about 8 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel sticky).
Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 1 hour and 15 minutes or until doubled in size. (Gently press two fingers into dough. If an indentation remains, the dough has risen enough.) Punch dough down; cover and let rest 5 minutes.
Turn dough out onto a lightly floured surface. Divide dough into 10 equal portions. Working with one portion at a time (cover remaining dough to prevent drying), shape each portion into a ball. Make a hole in the center of each ball using your index finger. Using fingers of both hands, gently pull dough away from center to make a 1 1/2-inch hole. Place bagels on a baking sheet coated with cooking spray.
Lightly coat bagels with cooking spray; cover with plastic wrap. Let rise 10 minutes (bagels will rise only slightly).
Preheat oven to 400°.
Combine remaining beer, 4 cups water, and sugar in a Dutch oven. Bring to a boil; reduce heat, and simmer. Gently lower 1 bagel into simmering beer mixture. Cook 30 seconds. Turn bagel with a slotted spoon; cook an additional 30 seconds. Transfer bagel to a wire rack lightly coated with cooking spray. Repeat procedure with remaining bagels.
Place bagels on a baking sheet sprinkled with cornmeal. Combine 1 teaspoon water and egg yolk in a small bowl; stir with a fork until blended. Brush bagels with yolk mixture; sprinkle with sesame and poppy seeds.
Bake at 400° for 17 minutes or until golden. Transfer to a wire rack to cool.