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Bring your eggs and butter to room temperature. I like to put the eggs in a bowl of warm water for 10 minutes and microwave the butter in 15-second increments until I can press my finger into the butter easily but it's still holding its shape and isn't melted.
Heat your milk to 80ºF-90ºF using a microwave or the stove
Add your instant yeast and bread flour and mix until just combined.
Cover the container and set it aside in a warm environment (roughly 75ºF) until it doubles in size (about 30 minutes). This gives the yeast a head start on growing.
Add your yeast mixture to the bowl of your stand mixer with the paddle attachment attached.
While mixing on low, add in the room temperature eggs one at a time, letting them incorporate before adding in the next.
Continue mixing on low and add in the bread flour, sugar, and then salt. Mix for five minutes.
Add the softened butter one chunk at a time, letting each piece incorporate before adding the next. The dough will be very sticky after adding in the butter. Be sure to scrape the bowl as needed to make sure the dough is mixing correctly.
Switch to the dough hook and let your dough mix on medium until a smooth dough forms that pulls away from the sides of the bowl. This can take anywhere from 10 minutes to 20 minutes depending on the mixer, the flour you're using and even the humidity in the air so be patient. It will be a very loose and sticky dough.
Form it into a smooth ball by gently folding the edges of the dough to the center with a bowl scraper.
Place the brioche dough into a buttered bowl, edges down, cover with a towel and let it rise in a warm spot for 30 minutes.
Cover the dough with plastic wrap and refrigerate the dough for two hours or overnight. The longer your dough chills, the easier it will be to handle and the more the flavor will develop. You can chill it up to 48 hours before shaping.
Once the dough is chilled, you can form it into three equal pieces. Roll the pieces out into long strands that are about 1 1/2 times as long as your pan.
Braid the three pieces together and pinch the ends together.
Tuck the ends underneath and transfer the dough to a 5"x9" ungreased non-stick bread pan. If you use any other non-stick pan, grease the pan first. You can also bake the loaf directly on a sheet pan with some parchment paper, in cupcake pans to make little braided rolls, or even a cake pan.
Cover the dough to prevent it from drying out and let the dough rise until it doubles in size at 75º-90ºF. This should only take 30 minutes to an hour depending on how warm the environment is. You will know the dough is ready if you gently poke it with your finger and it makes an indent that either stays or only bounces back halfway. If it bounces back immediately then it needs more time to rise.
Preheat your oven to 375ºF and place the oven rack in the lower half. If your bread is too close to the top of the oven it will get too brown.
Brush the surface of the loaf with egg wash (one egg whisked together with 1 Tablespoon water) so that it is nice and shiny when it comes out of the oven.
Bake the brioche until it's golden brown and the internal temperature reads between 180ºF-190ºF. Between 30-45 minutes, start checking the internal temp at 30 minutes.
Remove the bread from the pan and transfer it to a cooling rack. Enjoy your brioche when it's still a little warm! So good!