For apple pie filling, combine sugar, corn starch and cinnamon in a large bowl. Add apples and lemon juice; toss to coat. Set aside.
For crumb topping, combine flour, brown sugar, butter and vanilla in a small bowl, using a fork or pastry blender to mix. Set aside
For dough, combine 1 cup flour, sugar, undissolved yeast and salt in a large mixer bowl. Heat water, milk and butter until very warm (120°F to 130°F). Gradually add to flour mixture. Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Add eggs and 1 cup flour; beat 2 minutes at high speed. Stir in enough remaining flour to make a soft dough. Knead on a lightly floured surface until smooth and elastic, about 8 to 10 minutes. Cover; let rest 10 minutes.
Divide dough in half; roll each half into a 14 x 12-inch rectangle on a lightly floured surface. Transfer to two large greased baking sheets. Spread half of apple pie filling over a 3-inch strip in center of each dough rectangle. Sprinkle each with half of crumb topping. With a sharp knife, make 3-inch cuts from dough edges at 1-inch intervals along sides of filling. Alternating sides, fold strips at an angle across filling. Cover; let rise in warm, draft-free place until doubled in size, about 1 hour.
For egg glaze, combine egg white and water; brush over braids. Bake in a preheated 350°F oven for 20 to 30 minutes or until light golden. Remove braids from baking sheets to cool on wire racks.
For icing, combine powdered sugar, milk, butter and vanilla in medium bowl. Stir until smooth. Drizzle icing over braids; serve warm or at room temperature.