Brown the butter: in a medium sized saucepan add the butter and set over medium heat. Cook and stir as the butter melts, then starts to separate (you’ll hear it sputtering and see tiny bits that look like panko at the bottom of the pan). Continue cooking and stirring frequently until the bits at the bottom all turn a toasty brown color. Immediately transfer to a heatproof bowl.
Before you begin making the dough, the butter should be at a soft solid, or creamy state. Chill it for about 30 minutes to get to that texture.
Make the roux: in a small pot, add the water and flour and set over medium heat. Cook while whisking and stirring until the mixture thickens into a paste, about 5-7 minutes. Set aside to cool.
Make the dough: Add the quarter cup of water to the bowl of your stand mixer and sprinkle the yeast over it. Over the yeast add a pinch of the sugar. Stir then leave it for a few minutes to activate (if it does not foam the yeast is likely expired).
Over the yeast add the flour, sugar, salt, eggs, roux, milk powder and vanilla (if the roux is hot, try to keep the eggs away from it by putting the roux in one side of the bowl and the eggs in the other). Affix the dough hook and begin kneading. Once the dough starts to come together around the hook, you can add the butter, slowly - about one tablespoon at a time.
Knead the dough until the dough comes together again around the hook - this can take up to 20 minutes with the full amount of dough in the mixer. The dough will look soupy and separate at first but as it kneads it will become glossy and almost clear the bowl so that it’s mostly around the hook (see photos & video for reference). If your mixer is overheating, turn it off and give it a 5-10 minute break then continue kneading.
Set the dough in a large oiled bowl, and toss the dough around the bowl to coat it with oil. Cover with plastic wrap and set at room temperature to rise until doubled, about 2 hours (in a cold kitchen this may take longer, in a warm kitchen it might be quicker).
Ready two loaf pans by greasing them with a light spray of oil or brushing them with butter.
On a floured surface divide the dough in half, each half will become one loaf.
Per loaf you can either divide the dough into thirds and braid them, or divide the dough into eighths. With each piece of dough, roll it out into a long oval then roll the oval up like you would a cinnamon roll. Stack the eight rolled doughs in the loaf pan and repeat the process with the second half of the dough.
Let the two loaves rise for about 45-60 minutes, until the dough doesn’t immediately spring back when pressed but leaves a slight indentation. Preheat the oven to 350 F and beat together the egg yolk and heavy cream. Brush over the tops of the dough.
Bake the loaves until a thermometer inserted into the center registers 190 F, about 45 minutes. Check them at 30 minutes, and if they are over browning, tent them with foil (I tend to tent mine at the 25 minute mark). Let cool briefly in the pan then transfer to a cooling rack.
Store brioche loaves in an airtight container, they’ll last about 3 days.