Adapted from King Arthur Flour Baking Companion
Yields one loaf, 12 slices
-2 1/2 cups unbleached all-purpose flour
-1 1/2 teaspoons instant yeast
-1/4 cup cool water (about 70 degrees F)
-4 large eggs, plus 1 yolk for glaze
-2 tablespoons sugar
-3/4 teaspoon sea salt
-1 cup (8 ounces) cold unsalted butter
Place 1 1/2 cups of the flour, the yeast, water, and whole eggs in a mixer bowl. Beat at medium speed, using the flat beater paddle, until smooth. Cover the mixture and let sit for 45 minutes. It will develop some bubbles, but not change a lot due to the thinness of the batter. However, the yeast is getting a head start.
Add the remaining flour, the sugar, and salt, and beat for 8 to 10 minutes, until the dough cleans the sides of the bowls and is shiny and elastic.
Sprinkle a work surface with a small palmful of flour (about 2 tablespoons of flour). Place the butter on the flour. Pound with the side of a rolling pin until it has become a cohesive rectangle about 1/4-inch thick. Fold it over several times as you pound; it will become pliable without getting too warm or soft. Add it to the dough and beat until it is fully incorporated.
Cover the dough with a towel and let it rise in a warm room for an hour. It should be very soft and have risen about 1/3. Turn the dough out on a lightly floured surface and fold it over several times. (Use a bench knife or dough scraper to scrape up any bits that stick to the table.) Place the dough in a greased bowl, cover the bowl, and refrigerate for at least 4 hours (or up to 16). The dough will firm up considerably and be easier to work with.
Working quickly, shape the dough in a 8 1/2? x 4 1/2? x 2 3/4? high loaf pan. Cover the dough lightly and let it rise for 2 1/2 to 3 hours, until it is doubled and is crowded well over the top of the pan.
Preheat the oven to 375 degrees F. Beat the egg yolk with 1 tablespoon of water and brush all exposed surfaces with the egg wash. (Try not to let any egg wash drip onto the edge of the pan as it will cause the bread to stick.) Cut four slashes in the top of the loaf.
Bake for 45 to 50 minutes, until its internal temperature reads 190 degrees F and the top is golden brown. If you like a lighter crust, and I do, tent the bread with aluminum foil after 20 minutes of baking.
Remove the brioche from the oven and cool it in the pan for 10 minutes. Turn it out of the pan and let it cool completely on a rack. Slice the brioche when it is completely cool.