The starter must be made 48-72 hours in advance. Combine yeast and 1 c tepid water. Beat in 1 c rye flour. Add onion, cover with plastic wrap and let stand at room temperature for 24 hours. Remove onion. Beat in remaining water and rye flour. Cover and let stand at room temperature another 24 hours. It may be used at this point, or refrigerated up to another 24 hours.
Combine 1/4 c warm water, yeast and sugar. Let stand 10 minutes until foamy.
In work bowl of a heavy-duty mixer, dissolve salt in remaining warm water. Stir down sourdough starter and remove 1/2 cup (see head note). Mix remaining starter into salt water, then add yeast mixture. Add high-gluten flour and 1 c all-purpose flour; if using, add caraway seeds. Mix to form a soft dough.
Add remaining flour and knead until dough is soft and slightly sticky (either in the mixer or by hand). Do not overknead; the dough should not be very elastic. Form into a ball and place in an ungreased bowl. Cover and let stand at room temperature until doubled in size, about 90 minutes.
Punch down, knead briefly and let stand 15 minutes. Form into a 12" oval, turning the edges in toward the center to form a smooth outer surface. Pinch the seam to seal and place seam-side down on a baking sheet coated with cornmeal. Cover with a damp towel and let rise until almost double, about 45 minutes.
Meanwhile, place quarry tiles or baking stone in top 1/3 of oven, and an empty roasting pan in the bottom third. Preheat oven to 400°F.
Beat egg white with water and brush over surface of loaf. Slash loaf with a very sharp knife 2 or 3 times. Sprinkle with additional caraway seeds if desired. Place loaf on tiles and toss 4 cups of ice cubes into the hot roasting pan. Bake 30 minutes. Brush loaf with more egg white glaze and bake another 20-30 minutes.