Warm the water to 115 F.
Place the bread flour in a large mixing bowl and add the salt and yeast. Pour in the honey and oil. Add eggs and egg yolk, setting the egg white aside.
Gradually add the warm water as you mix on low speed. The dough should be soft and slightly sticky. Knead on medium-low speed with a dough hook for 5-7 minutes, until the dough is smooth, no longer sticky, and has a glossy sheen on its surface.
Shape the dough into a ball, place in an oiled bowl, and cover tightly. Allow to rise in a warm place until doubled in size.
Turn out the dough onto a lightly floured surface and punch it down. Divide into 5 equal pieces. Roll each piece into a rope about 20 inches long. Line the ropes up side by side, then squeeze them together at the top to join. Split the strands into two groups: a group of two on the left and a group of three on the right.
Begin the braiding by crossing over the third strand from the second group and placing it beside the second strand in the first group. Then, cross over the first strand from the first group and place it beside the first strand in the second group. Repeat this process until the whole challah is braided and pinch the end to seal. Place it on a lightly floured cookie sheet.
Cover and let rise in a warm place until doubled in size, about 1 hour.
Preheat the oven to 375 F.
Beat the egg white with a little water to thin it, then brush it on the risen loaf. Bake at 375 F. for 10 minutes. Lower the temperature to 325 F. and bake for another 15-20 minutes. The bread should be a rich golden brown and have an internal temperature of 190 F.
Let the loaf cool completely on a wire rack before slicing and serving. Serve with salt or butter.