Combine milk sugar and yeast in a large bowl; let stand 5 minutes or until bubbly. Stir in butter, salt, 5 egg yolks, and 3 eggs. Stir in fontina and chives. Weigh or lightly spoon flours into dry measuring cups; level with a knife. Add bread flour and all-purpose flour to yeast mixture, stirring until a soft dough forms (dough will be sticky).
Turn dough out onto a lightly floured surface. Knead until smooth and elastic, adding remaining 1/4 cup all-purpose flour, 1 tablespoon at a time, to prevent dough from sticking to hands. Place dough in a large bowl coated with cooking spray, turning to coat top. Cover with plastic wrap, and let rise in a warm place (85°), free from drafts, for 45 minutes or until doubled in size. Punch down dough; cover and let rise 50 minutes or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.)
Divide dough into 6 equal portions. Roll each portion into a ball. Roll each ball into a rope about 15 inches long. Place 3 ropes parallel to one another; braid ropes. Pinch ends together, and tuck under loaf. Repeat procedure with remaining 3 ropes. Place loaves on a baking sheet lined with parchment paper; coat with cooking spray. Cover and let rise 30 minutes or until doubled in size.
Preheat oven to 375°.
Combine 1 egg and 2 tablespoons water, stirring well with a whisk. Brush loaves gently with egg mixture. Sprinkle loaves evenly with Parmigiano-Reggiano.
Bake at 375° for 25 minutes or until golden. Remove from baking sheet; cool on a wire rack.