Instructions

1. In a stand mixer fitted with the paddle attachment, mix the flour, sugar, yeast, and salt on low speed until well combined. Add the eggs and the milk and mix on low speed. When the dough starts to clump together, remove the paddle attachment and attach the dough hook. Mix on medium speed for 2 minutes until it is firm and elastic, about 2 minutes more.
2. With the mixer on medium-low speed, add the butter, a few pieces at a time. Once all of the butter has been added, increase the mixer speed to medium and mix for 4 minutes. Scrape the dough hook and the sides and bottom of the bowl. Mix again until the dough is smooth, soft, and shiny, about 4 minutes more. 
3. Transfer the dough to a lightly floured surface. Knead it by hand a few times and then form it into a ball. Cover loosely with plastic and let the dough rise in a warm spot until doubled in size, about 1 hour.
4. Repeat step 3 and let the dough rise again. Cover tightly with plastic. (At this point, you may refrigerate the dough overnight for the best results.) Or let it sit out until doubled in size, about 1 hour. The warmer the room, the faster the brioche will rise, so keep an eye on it.
5. Pre-heat the oven to 375°F (190°C) and butter a regular muffin pan. (If the dough was refrigerated, let it warm to room temperature, about 2 hours.) Turn the dough out, smooth top down, onto a clean work surface. Divide the dough into 12 equal pieces. Cover the dough to prevent it from drying out. Roll each piece of dough into a tight ball, and then into a rod and moving it in a circular motion to form the shape above.
6. In a small bowl, make the egg wash by beating the remaining egg and a pinch of salt. Lightly brush the top of the brioches. Bake until golden-brown, or about 18 minutes. Let cool before turning them out. Serve warm.