Combine 1 cup flour, sugar, yeast, and salt in a large bowl.
Heat milk, 1/4 cup water, and 3 tablespoons butter in a saucepan until very warm (120° to 130°); gradually stir into flour mixture. Stir in 1 egg and remaining 1 1/2 cups flour until blended.
Turn dough out onto a floured surface, and knead until smooth and elastic (4 to 6 minutes). Place in a well-greased bowl, turning to grease top. Cover and let stand 10 minutes.
Combine onion and next 3 ingredients in a small bowl.
Roll dough into a 14- x 10-inch rectangle; cut in half lengthwise. Spoon half of onion mixture down center of each rectangle. Bring long sides over filling, pinching seams to seal. Place, seam side down and side by side, on a lightly greased baking sheet. Pinch portions together at 1 end to seal; braid portions, and pinch ends to seal.
Cover and let rise in a warm place (85°), free from drafts, 20 to 30 minutes or until doubled in bulk.
Stir remaining 1 egg and 1 tablespoon water until blended; brush on dough. Sprinkle with poppy seeds.
Bake at 350° for 35 minutes, shielding with aluminum foil after 25 minutes; cool on a wire rack.