Combine 2 cups flour, undissolved yeast, sugar and salt in large mixer bowl. Heat water, milk and 1/4 cup butter until very warm (120° to 130°F). Add to flour mixture. Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally.
Stir in enough remaining flour to make a soft dough. Knead on lightly floured surface until smooth and elastic, about 8 to 10 minutes. Cover; let rest 10 minutes.
Divide dough into 8 equal pieces; roll each piece to a 6-inch rope. Shape into oval roll. Place on large greased baking sheet. Place 3-inches apart for crisper buns. Place 1/2-inch apart for softer buns (more like commercial hot dog buns). Cover; let rise in warm, draft-free place until doubled in size, about 30 to 45 minutes.
Brush buns with egg white and water mixture and sprinkle evenly with poppy seeds.
Bake in preheated 375°F oven for 15 to 20 minutes or until done. Remove from sheet; cool on wire rack.
Split top of bun and add hot dog. Pile on all your favorite Chicago dog toppings!