Start by letting your mixing bowl get warm by running it under hot water for a minute and then drying it off before adding anything into it.
Dump the cup of hot water, sugar, and yeast into the bowl and mix them up by hand and let sit in a warm place for 10 minutes until foamy. (This activates the yeast.)
In another bowl, mix the whole 2 eggs and the 2 egg yolks with salt an honey until well-incorporated and let sit while the yeast gets going. Save the egg whites for the egg wash.
After 10 minutes, add the melted butter to the yeast mixture and stir by hand. Then add the honey-egg mixture and give it a good stir by hand until you feel no honey stuck to the bottom of the bowl.
This is where the stand mixer comes in handy. Attach the bowl to the stand and add 1 cup of bread flour and stir on medium-low with the paddle attachment until smooth-ish. Repeat with 1 cup of all-purpose flour, again with bread flour, and one more time with all-purpose flour.
After all the flour has been mixed in, scrape the bowl with a spatula and let it go on medium for about a minute. It should be sticky and the dough should be loose and feathery. If it is too runny (depending on climate), add ¼ cup more of all-purpose flour and mix one more time and you should be good to go.
Microwave a large glass mixing bowl and coat completely with vegetable oil, then add the dough and toss it around a couple times until it's also coated with oil. Loosely cover with a plastic wrap and let sit in a warm, draft-free place. (I like to use a lightly warmed oven. I'd turn it on the warm setting for a few minutes and then turn it off before putting the bowl in there.) Now, let the dough sit in there for 1 hour, no less.
Then take it out, punch it down, and dump onto a clean, floured work surface to braid. I like to toss it in the flour a little to make it a little less sticky before attempting to braid, it makes a smaller mess. After getting it nicely coated with flour, cut into 3 equal parts for one loaf, or 6 equal parts for 2 loaves. Then roll out to about a 2-inch diameter and braid each. Once braided, pinch and tuck the ends under to make it pretty.
Now put it onto whatever pan you're using to make it. A half sheet pan works nicely for a full loaf or baking both small loaves at once. Put a piece of parchment paper on the pan (very necessary, trust me) and put the bread in the middle. Put it back into the oven for another 45 minutes to an hour to rise.
Remove the bread and heat the oven to 350°. Once heated, bake the bread on the middle/top rack. Here's a little trick of mine: The bread bakes for 30 minutes in total, however, I do not apply the egg wash until it has baked for 10 to 12 minutes. This keeps it from getting too brown. Another thing, I do right when it hits the color I want (which is about 20 minutes. I rotate the pan and cover loosely with a foil tent. This stops the browning but allows the bread to remain nice, shiny, and beautiful. After 30 minutes, take it out and remove from the pan, but leave it on the parchment paper for 10 minutes before you tear, slice, or anything. It keeps the bread more moist. Enjoy!