Combine 3 cups all-purpose flour, sugar, salt, and yeast in a large mixing bowl; stir well. Combine water, saffron, and shortening; heat in a saucepan until mixture reaches 120° (shortening does not need to melt completely).
Add liquid mixture to flour mixture, beating well at medium speed with a heavy-duty mixer using the paddle attachment. Add 1 egg and 1 cup flour. Beat at low speed 1 minute; then beat at medium-high speed for 3 minutes. Stir in enough remaining flour to make a soft dough.
Turn dough out onto a lightly floured surface, and knead until smooth and elastic (about 10 minutes). Shape dough into a ball; place in a well-greased bowl, turning to grease top. Cover and let rise in a warm place (85°), free from drafts, 45 minutes or until doubled in bulk.
Punch dough down, and divide in half. Set half of dough aside. (Cover to prevent drying.)
Divide remaining half of dough into 4 equal portions. Combine 3 portions; knead 2 or 3 times on a lightly floured surface. Cover and set aside remaining portion of dough. Roll larger portion into a 13" x 10" rectangle; transfer to a lightly greased baking sheet.
Spoon half of Cheddar-Apple filling in a 4" wide strip down the center of dough, leaving 3" on each side. With a sharp knife, slit dough at 1" intervals along each side of filling. Fold strips over filling, alternating from side to side, tucking ends in.
Divide reserved smaller portion of dough into thirds. Shape each third into a 14" rope; pinch ends together at one end to seal. Braid ropes; pinch loose ends to seal.
Brush top of apple-filled braid lightly with 1 beaten egg; place smaller braid across top of larger braid. Brush smaller braid with egg. Tuck ends of smaller braid under bottom edge of larger braid, pinching to seal. (Place a wooden pick through braids at both ends to prevent separation during rising and baking.)
Repeat rolling, cutting, filling, and braiding procedures with remaining half of dough and filling. Cover and let loaves rise in a warm place, free from drafts, 45 minutes or until doubled in bulk.
Bake at 350° for 25 to 30 minutes or until lightly browned. Transfer to a wire rack; remove wooden picks, and cool. Brush with melted apple jelly; cool. Sift powdered sugar over loaves before serving.