Preheat the oven to 375° F. Line a baking sheet with parchment paper.
In the bowl of an electric mixer fitted with the dough hook attachment, mix the flour, sugar, yeast, and salt to combine.
Add the liquid ingredients (water, milk, honey, oil, and eggs), and continue to mix on low speed for 3 minutes. Raise the speed to medium and continue to mix for 5 minutes more.
Transfer the dough to a large bowl lightly greased with non-stick spray. Cover loosely with plastic wrap and let rise at room temperature until double in size, about 1 hour.
To braid the bread, turn out the dough onto a lightly floured surface. Divide it into three even pieces. Roll each piece into a log about 12 inches long. Pinch each piece together at the top to seal, leaving the bottom ends loose.
Braid the bread just as you would strands of hair: Move the right hand piece to the middle, then the left hand piece to the middle -- and so on and so forth until all of the dough is braided.
Transfer the loaf to a the prepared baking sheet, and cover loosely with plastic wrap. Let rise for 30 to 45 minutes more.
Brush the egg wash onto the loaf just before baking. Bake the loaf until deeply golden brown and fully baked through, 40 to 50 minutes. Cool before slicing and serving.
I always carry three kinds of hot sauce in my purse. I have a soft spot for making people their favorite dessert, especially if it's pie. My first cookbook, The Fearless Baker, is out on October 24, 2017.