In stand mixer fitted with paddle attachment, add 3 cups flour, yeast, sugar, salt and onion; mix well. Add warm water and shortening to flour mixture; blend on low speed for 30 seconds. Add 2 eggs; beat on medium speed for 3 minutes.
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 30 minutes.
Prepare pans: Grease two 9x5-inch bread pans; set aside.
Punch down dough. Divide into 2 parts. On lightly floured surface, roll or pat each half to a 14x7-inch rectangle. Starting with shorter side, roll up tightly, pressing dough into roll with each turn. Pinch edges and ends to seal. Place in prepared pans, seam side down. Cover; let rise until indentation remains after touching, about 40 minutes.
Preheat oven to 375ºF .
Brush loaves with egg wash. With sharp knife, carefully make a slash 1/2-inch deep lengthwise down center of each loaf. Bake for 35 to 40 minutes until golden brown. Remove from pans; cool on wire rack.