Brioche

In the bowl of a stand mixer fitted with the dough hook attachment, add flour, salt, and yeast, making sure to place yeast and salt apart from one another in the bowl. Add milk, 3 eggs, and the egg yolk and mix on low speed until no dry flour remains, about 1 minute. Increase speed to medium and mix for 3 minutes, then let rest for 3 minutes. Repeat mixing on medium speed and resting dough, scraping down bowl and hook as needed, until dough is homogenous and somewhat elastic (you can stretch it but the dough will tear fairly easily).

With the mixer running on medium speed, add one-third of the butter, one piece at a time, making sure to wait until each piece is mostly incorporated before adding the next, about 4 minutes. Add sugar and mix until fully incorporated, about 1 minute (the dough will slacken and look more wet). Repeat adding remaining butter, one piece at a time, and scraping down bowl and hook as needed, about 8 minutes. Let dough rest for 2 minutes.

Increase speed to medium-high and mix until dough pulls away cleanly from sides of bowl and is able to be gently stretched into a thin sheet without tearing, about 8 minutes.

Scrape dough onto a clean, un-floured work surface. Using clean hands, slide hands under left and right side of dough and lift it up from the middle until dough releases from work surface, then slap the bottom half of the dough down onto the work surface, quickly stretching the top half up and folding it over the bottom half (it will be sticky at first but don't worry, this process will reduce that as you work; do not add flour). Repeat this slap and fold motion , rotating dough as you work, until dough is smooth and less sticky, about 5 to 10 slaps.

Transfer dough to a greased parchment-lined rimmed baking sheet, cover with plastic wrap, and let dough proof at room temperature until dough is visibly puffed, about 1 hour. Press dough into a roughly 8- by 14-inch rectangle, cover tightly with plastic, and place in the refrigerator. Let dough proof and allow the butter to solidify for at least 4 hours or up to 24 hours.

Grease an 8 1/2- by 4 1/2-inch aluminum loaf pan and line with a 7- by 13-inch sheet of parchment paper so that parchment hangs over the long sides of the pan to form a sling. Transfer chilled dough onto a lightly floured work surface. Using a rolling pin or your hands, stretch the dough into a 8- by 14-inch rectangle. Starting at the narrow end, roll dough to form a log, then transfer to the prepared loaf pan seam side down. Alternatively, for brioche Nanterre, divide dough into 8 equal portions (70g each), forming each into a smooth ball. Arrange dough in a single layer in prepared loaf pan; they can be slightly touching and will proof into each other.

Loosely drape plastic wrap over pan to prevent a crust from forming and let dough proof at room temperature, between 2 1/2 to 5 hours for a large loaf, or 1 1/2 to 3 hours for a Nanterre-style loaf. The loaf is ready to bake once the dough is peaking above the rim of the pan by 1 inch and looks very wobbly. When pressed with a lightly wet or oiled finger, your finger should leave a slight indentation and the dough should bounce back slowly.

Adjust oven rack to middle position and preheat to 400°F (204°C). In a small bowl, whisk remaining egg until homogenous, then using a pastry brush, gently brush all over loaf.

Place loaf in the oven, reduce temperature to 350°F (180°C), and bake until deep golden brown and to an internal temperature of 190°F (88°C), about 35 minutes (if loaf is darkening too quickly, loosely cover with aluminum foil); if loaf has not reached desired internal temperature, continue to bake, checking temperature in 5-minute increments until desired final temperature is reached.

Transfer pan to a wire rack and let cool for 10 minutes. Run a butter knife along edges to loosen, then lift loaf using overhanging parchment. Remove parchment, return loaf to wire rack, and let cool completely. Slice and serve as desired.