Place the warm water and honey in a large mixing bowl and sprinkle the yeast on top.
Stir in the yeast until it's dissolved, and set it aside until small bubbles start to form.
When the mixture is bubbly or foamy on top,*** stir in the egg, egg yolk, and oil.
Add about half the flour, mixing on medium-low speed until incorporated.
While continuing to mix, add small amounts of flour, just until the dough gathers itself into a ball and pulls cleanly away from the sides of the bowl. (You may need more or less flour depending on environmental conditions; use your best judgement.)
Mix in the salt.
Knead the dough (by hand or by machine) until it passes the windowpane test**** (about 5 to 10 minutes).
Place the dough in a greased bowl, cover with plastic wrap, and allow to rest in a warm place until doubled in volume (about 1 to 2 hours).
When the dough has grown to twice its original size, punch it down, knead it a few times, and divide it into 4 equal portions.
Roll each portion of dough into a long rope, and pinch all 4 ropes together at the top.
Starting with the left-most rope, weave it over and under each of the other ropes, moving across to the right. (See video.)
Repeat this step until the end of the braid has been reached, then pinch together at the bottom.
Transfer the braid to a parchment-lined baking sheet, brush with egg wash, and cover loosely with greased plastic wrap.
Allow the loaf to rise a second time,***** until doubled in bulk (1 to 2 hours).
Preheat the oven to 350 degrees and brush the loaf with egg wash a second time.
Bake the challah bread for 20 to 30 minutes, or until puffed and deeply golden.