Heat oil in a large skillet over medium-high heat. Add onion; sauté 4 minutes or until golden brown. Cool.
Dissolve 1/8 teaspoon sugar and yeast in warm beer; let stand 5 minutes. Stir in yogurt, vinegar, caraway seeds, and salt. Add remaining 1 teaspoon sugar and egg; stir with a whisk until combined.
Lightly spoon flours into dry measuring cups; level with a knife. Add 2 1/2 cups all-purpose flour and rye flour to yeast mixture; stir until a soft dough forms. Stir in onion mixture. 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 tacky).
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, 45 minutes or until doubled in size. (Press two fingers into dough. If indentation remains, the dough has risen enough.) Punch dough down; cover and let rest 5 minutes.
Shape dough into a 12-inch oval loaf on a lightly floured surface. Place loaf on a baking sheet lined with parchment paper. Lightly coat surface of loaf with cooking spray. Cover with plastic wrap; let rise 30 minutes or until doubled in size.
Preheat oven to 400°.
Combine 1 teaspoon water and egg white in a small bowl. Gently brush egg white mixture over surface of loaf. Bake at 400° for 28 minutes or until loaf is golden brown and sounds hollow when tapped, and let cool on a wire rack.