TO MAKE THE DOUGH: In the bowl of a stand mixer with paddle attachment, mix together Bread Mix, sugar, and yeast. Add eggs, milk, warm water, and melted butter and mix on low until combined, then increase the speed to medium and beat for three minutes.
Label three pastry piping bags or three large ziplock bags Bag #1, Bag #2, and Bag #3. Divide the bread dough evenly among the bags.
TO BRAID THE DOUGH: Because this dough is too soft to actually braid, you will pipe the dough out like frosting, one small section at a time, to form the braid.
Set out a 13 x 18-inch sheet of parchment paper. Using a dark marker or pen, draw two lines from top to bottom of parchment paper, 4 inches apart, so the paper is divided into 3 equal columns. You will pipe the braid within the middle column. Turn parchment paper over so line is facing down and put into 13 x 18-inch sheet pan.
Cut tips of piping or ziplock bags so openings are about 1-inch wide.
Using Bag #1, start braid at the top of the pan with tip in the middle of the column; squeeze dough out diagonally downward to the right until you reach the right edge of the column. Stop there and set the bag down, without breaking off dough, with the tail of the bag laying outside the column to the right.
Take Bag #2 and point tip just to the right of where you started the braid with Bag #1. Squeeze dough out, this time pulling downward diagonally to the left, crossing over the top of strand #1, until you reach the left side of the column; set bag down without breaking off dough, with the tail of the bag laying outside the column to the left.
Now take Bag #3 and point tip just to the left of where you started the braid with Bag #1. Squeeze dough out downward diagonally to the right, crossing over strand #2, until you reach the right side of the column. Stop there and set the bag down without breaking off dough, with the tail of the bag laying outside the column to the right, below Bag #1.
Pick up Bag #1 again and squeeze dough out downward diagonally to the left over strand #3 until you reach the left side of the column. Pick up Bag #2 and continue the pattern with all three bags to form the braid until you run out of dough. Tuck the ends of the braid neatly under the loaf. Smooth the ends with damp fingers. At this point, dough may be frozen to be baked later (see note below).
Let challah rise for 45 to 60 minutes in a warm, draft-free spot until almost doubled in size.
Preheat oven to 350°. Mix egg yolk with 1 tablespoon water to make egg wash. Brush loaf gently with egg wash before baking. Cover loosely with foil so bread does not get too brown. Bake for about 60 minutes.
Let cool completely before slicing with a serrated knife.
To freeze challah: Spray challah with nonstick cooking spray before it has risen, leave uncovered on sheet pan, and put sheet pan in freezer until challah is hard enough to pick up without losing its shape. Remove challah from freezer, wrap tightly in plastic wrap, and return to freezer until ready to use.
To bake challah from frozen: When ready to use, remove plastic wrap from challah to prevent condensation while thawing. Loosely cover with plastic wrap or a cover to keep dough from drying out and let thaw in refrigerator overnight. The next day, remove challah from refrigerator and allow it to come to room temperature. Let challah rise for about 60 minutes in a warm, draft-free spot until almost doubled in size. Bake as directed above.
To make with bread machine: Dough can be made with bread machine following bread machine instructions, mixing dough up to the first rising.
If you need to see how a braid is formed, you can make a practice braid out of ribbon or string to look at while you pipe the dough.