Combine first 6 ingredients in a heavy saucepan over low heat. Cook 5 minutes or until sugar dissolves and a thermometer registers 115°, stirring frequently. Remove saucepan from heat.
Lightly spoon flour into dry measuring cups; level with a knife. Combine 2 1/2 cups flour and yeast in a large bowl, stirring well with a whisk. Add milk mixture and egg; beat with a heavy-duty stand mixer at low speed 30 seconds or just until blended. Increase mixer to medium speed; beat 2 to 3 minutes. Gradually add 2 cups flour; beat at low speed until mixture forms a ball.
Turn dough out onto a floured surface. Knead until smooth and elastic (about 8 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel sticky). Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in size. (Press two fingers into dough. If indentation remains, dough has risen enough.) Punch dough down; cover and let rest 5 minutes.
Preheat oven to 375°.
Divide dough into 3 equal portions, shaping each into a 20-inch rope. Pinch ends together at one end to seal. Braid ropes on a lightly floured surface; pinch loose ends to seal. Place braid on a baking sheet coated with cooking spray. (The braid should be about 15 inches long.) Cover and let rise 30 minutes or until doubled in size.
Combine egg white and water; brush over top of loaf. Sprinkle 1 tablespoon coarse-grain sugar over loaf. Bake at 375° for 25 minutes or until golden and loaf sounds hollow when tapped. Remove from oven; cool on a wire rack.