Heat vegetable oil in a large skillet over medium heat. Add onion; cover and cook 10 minutes or until golden brown, stirring frequently. Remove from heat; cool completely.
Dissolve sugar and yeast in warm water in large bowl; let stand 5 minutes. Stir in yogurt, caraway seeds, salt, and pepper. Add egg; stir well with a whisk.
Lightly spoon flours into dry measuring cups, and level with a knife. Add 2 cups bread flour and rye flour to yeast mixture, 1 cup at a time, stirring until a soft dough forms. Stir in chopped onion. Turn the dough out onto a lightly 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. (Gently press two fingers into dough. If indentation remains, the dough has risen enough.) Punch dough down; cover and let rest 5 minutes.
With floured hands, knead dough 5 times. Shape into a round 7-inch loaf. Place loaf on a large baking sheet covered with parchment paper. Cover and let rise 30 minutes or until doubled in size. Make 3 diagonal cuts 1/4-inch deep across top of loaf using a sharp knife.
Preheat oven to 350°.
Combine 1 tablespoon water and egg yolk; gently brush over dough. Sprinkle with 1/2 teaspoon kosher salt. Bake at 350° for 35 minutes or until loaf sounds hollow when tapped. Cool on a wire rack.