In large bowl, combine 1 cup of the flour and the yeast; set aside. In small saucepan, combine the 1/3 cup butter, 1/4 cup milk, 2 tablespoons sugar, 2 tablespoons water, and the salt. Cook and stir over medium heat just until warm (120 degrees F to 130 degrees F) and butter almost melts. Add milk mixture to flour mixture along with the 2 eggs. Beat with electric mixer on low to medium speed for 30 seconds, scraping side of bowl constantly. Beat on high speed for 3 minutes. Using a wooden spoon, stir in as much of the remaining flour as you can.
Turn dough out onto lightly floured surface. Knead in enough of the remaining flour to make a moderately soft dough that is smooth and elastic (3 to 5 minutes total). Shape dough into ball. Place dough in lightly greased bowl, turning once to grease surface. Cover; let rise in warm place until double in size (1 to 1-1/4 hours).
Meanwhile, for filling, in medium bowl, stir together walnuts, pecans, beaten egg yolk, 1/4 cup milk, 2 tablespoons sugar, the honey, and the 1 tablespoon melted butter. Cover; set aside.
Punch dough down. Turn dough out onto lightly floured surface. Divide dough in half. Cover; let rest for 10 minutes. Meanwhile, lightly grease large baking sheet.
Roll each half of dough into 16x12-inch rectangle. Spread half of the filling over each portion of dough to within 1 inch of the edges. Roll each portion of dough into a spiral, starting from a long side. Brush edges lightly with water; seal seams and ends. Place loaves, seam sides down, on prepared baking sheet. Cover; let rise in warm place until nearly double in size (about 45 minutes).
Brush with egg white. Bake in a 350 degree F oven for 20 to 25 minutes or until golden and bread sounds hollow when lightly tapped. Remove from baking sheet. Cool on wire rack.
Combine powdered sugar and enough water (2 to 4 teaspoons) to make icing of drizzling consistency. Drizzle over loaves. Makes 2 loaves (32 servings).