In the bowl of a stand mixer fitted with the dough hook, mix the eggs, egg yolk, melted butter, and water together. Add the remaining dough ingredients and knead at low speed for about 5 minutes, until the dough is tacky but not sticky (you can add a couple of more tablespoons of flour if you really need to, but don't over-flour it)

While it kneads, whisk together the egg white and water for the egg wash and refrigerate.

Spray a large mixing bowl with cooking spray, then put the dough in it and cover with plastic wrap or a kitchen towel. Let rise in a warm place (I often turn my oven on for about 90 seconds, then turn it back off and let my dough rise in there) until doubled, about 1½-2 hours.

Press the dough down carefully to deflate it, then cover and let it rise again until its doubled in size, another 45-60 minutes.

On a lightly floured surface, divide the dough into three equal strands (like snakes - long and thin), then carefully braid them together (I generally start about 1/3 from the top, braid to the end, then braid the other end). Tuck the ends under.

Transfer to a baking sheet (I use my silicone baking mat to line it, but you could do no lining or use parchment paper instead).

Cover loosely with a towel or plastic wrap and let rise in a warm place for 30-45 minutes, until it has increased in size by about a third.

Preheat the oven to 375˚ F. Brush the loaf with the egg wash (I never use it all, so I just add it to my french toast liquid a day or two later). Bake the loaf for 25-35 minutes, or until it is golden brown. Cool before slicing (but I would never expect you to wait until it was COMPLETELY cool).