Sauté onion in 1 tablespoon hot oil in a small skillet over medium-high heat 4 to 5 minutes or until golden. Set aside.
Combine remaining 1 tablespoon oil, milk, and 1/4 cup water in a saucepan; heat to 120° to 130°.
Stir together 3/4 cup bread flour, 1/4 cup rye flour, sugar, salt, and yeast in a large mixing bowl; make a well in center of mixture. Stir in milk mixture with a wooden spoon. Stir in remaining 1/2 cup rye flour, onion, and caraway seeds. Gradually stir in enough remaining bread flour to make a stiff dough.
Turn dough out onto a floured surface, and knead until smooth (about 8 minutes). Place in a well-greased bowl, turning to grease top.
Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk.
Punch dough down, and turn out onto a lightly floured surface; let stand 5 minutes. Shape into a loaf, and place in a greased 9- x 5-inch loafpan.
Cover and let rise in a warm place, free from drafts, 45 minutes or until doubled in bulk. Stir together egg and 1 teaspoon water; brush over loaf.
Bake at 375° for 30 minutes or until loaf sounds hollow when tapped. Remove from pan immediately, and cool on a wire rack.