• Mix the dough. Combine flour, salt, and yeast the bowl of a stand mixer fitted with a dough hook. In a separate bowl, whisk together warm milk, honey, and egg. Pour the wet ingredients into the dry ingredients. Mix on low speed (KitchenAid speed 2-3) until the dough comes together in a shaggy messy ball on the dough hook, about 3-5 minutes. The dough will look dry at first, but will hydrate as it mixes. Be patient!
• Knead the dough. Increase speed to medium (KitchenAid speed 4) and knead until the dough passes the windowpane test, about 7-10 minutes. If the dough hasn't reached windowpane after about 7 minutes, drizzle in an additional ½ teaspoon milk while kneading, then cover and rest for 5-10 minutes. Knead 2-3 minutes more. It should get there!
If you don't reach a perfect windowpane before adding the butter and it's been at least 10 minutes of kneading, that's okay — you can go ahead and start adding butter. It won't be a technically perfect brioche, but it will still be delicious!
• Add the butter 1 tablespoon at a time. With the mixer running on medium (KitchenAid speed 4), add the butter one tablespoon at a time. Let each piece fully incorporate before adding more. The dough will look like it is falling apart each time you add more butter, but it will come back together. Pause the mixer to gather the dough around the hook or scrape the butter down into the bowl occasionally as needed. This can take 10-15 minutes. Be patient!
Each time you add a new piece of butter, the butter will smear all over the walls of the bowl and coat the outside of the dough. Your dough may fall apart a little bit and spit the butter out as it slides around in the bowl. The brioche dough might even fall off the dough hook completely. Just let the mixer keep running (or pause the mixer, gather the dough back into a ball, and then keep going) and it will come back together.
You're asking the gluten network to incorporate quite a lot of fat each time you add a new chunk of butter — be patient! At the end of each mixing stage, the dough will be slightly tacky to the touch, but should clear the sides of the bowl.
If the butter chunks are really struggling to incorporate, lightly dust in ¼ teaspoon of flour — it will help the butter cling to the dough. Resist adding more flour unless absolutely needed!
• Knead the dough again. Increase speed to medium-high (KitchenAid speed 5-6) and knead until the dough is smooth, shiny, and passes the windowpane test again, about 5 minutes. If the dough is super sticky, dust in more flour. If the dough is super dry, you may need to drizzle in a little more milk.
• Cover and rise. Place the dough ball in a lightly greased bowl or container. Cover and let rise 1 hour at room temperature (72-75°F) until just about doubled in size. If not doubled after an hour, let it rise an additional 30-60 minutes until doubled. When you push a finger into it, the indentation should fill back slowly and incompletely (the fingerprint test).
Kitchen temperature makes a big difference in how quickly your brioche rises. If your kitchen is cold (below 70°F), the butter will be more solid and the dough will rise slowly. If your kitchen is warm (above 75°F), the butter will be very warm and the dough will rise quickly. It's more important that the dough looks and feels right than that a certain amount of time has passed. If the dough is ready a little before an hour or a little after an hour, or if the dough needs another 30 minutes to rise, that's just fine.

Assembly and Shaping

• Deflate the dough. Turn dough out onto a clean, lightly floured work surface. Use your hands to gently deflate the dough. Divide the dough into eight equal pieces using a kitchen scale (each piece should weigh about 84 grams).
• Shape the brioche loaf. Gently flatten each piece of dough against the counter, stacking any smaller pieces on top of larger pieces if you're combining them. Fold or tuck the edges up across the middle of the dough to create a ball, then pinch the edges together to to create tension and a smooth round top on the other side of the dough. Place the dough down with the smooth side up and the pinched together seam on the counter. Cup your hand around it with your pinkie against the counter and scoot it in circles to tighten the seam and further tighten up the top.

  Repeat with the remaining 7 pieces of dough, then arrange in a greased and parchment lined loaf pan in two rows of four.
• Final Rise. Cover and let rise at room temperature for 1-2 hours or until the dough balls have doubled in size (see notes). When you gently poke the top of the dough it should feel soft and springy and the indentation should fill back in slowly but not completely.
How to tell when the loaf has doubled in size: If using a 9x5" loaf pan, the shaped brioche loaf will just reach the rim of the pan. If using an 8x4" loaf pan, the shaped brioche loaf should be just cresting over the rim of the pan. 
BTW: Sometimes brioche will develop air bubbles in the outer layer of dough. I usually gently pop these before egg washing. The egg wash seals them flat for a nice shiny top.

Baking

• During the last 30 minutes of the dough rising, preheat the oven to 350°F. Whisk together egg, milk, and salt to make the egg wash.
• Egg wash and bake. Brush the top of the dough with egg wash. Then bake for 30-35 minutes until golden brown on top and an internal temperature of at least 190°F. Check the loaf after 25 minutes; If the top appears to be browning too much, tent a sheet of aluminum foil over the top.
• Cool. Let cool slightly in the pan, then remove to a rack to finish cooling completely.