In a stand mixer bowl, combine 2 cups flour, yeast, sugar and salt; mix well. In saucepan or microwave-safe dish, heat water, milk and shortening/butter until warm (120-130°F; shortening/butter does not need to melt). Add to flour mixture. Blend at low speed, slowly adding slightly beaten eggs - one at at time, until fully incorporated; beat 3 minutes at medium speed.
Switch to dough hook attachment. Gradually stir in enough remaining flour to make a firm dough. Knead until smooth and elastic, about 5 to 8 minutes.(NOTE: For hand mixing: following above steps, mix ingredients in large mixing bowl using a wooden spoon or dough whisk. Knead on floured surface until smooth and elastic.)
Place dough in lightly oiled bowl and turn to grease top. Cover; let rise until indentation remains after poking dough with finger down to second knuckle, about 1 hour.
Punch down dough. Divide into 3 parts. On lightly floured surface, roll each third to a 15-inch rope. On greased cookie sheet loosely braid from center to ends. Pinch ends and tuck under to seal. Cover; let rise in warm place until indentation remains after lightly touching, about 30-45 minutes.
Preheat oven to 400°F.
Brush with egg wash. Bake for 25 to 30 minutes until golden brown. Remove from cookie sheet; cool on wire rack.