How to Make It
In a small bowl, whisk the yeast with the buttermilk until it dissolves. Let stand for 10 minutes, until foamy.
In the bowl of a stand mixer fitted with the dough hook, mix the bread flour with the sugar and salt. With the machine at medium speed, add the yeast mixture, then add 4 of the eggs, 1 at a time, beating well after each addition. Drizzle in the butter and beat for 10 minutes; the dough will look slightly greasy. Transfer to an oiled bowl, cover with plastic wrap and refrigerate overnight. Let the dough stand at room temperature for 1 hour before proceeding.
Lightly oil a 10-by-5-inch loaf pan. On a work surface, roll out the dough to a 10-by-8-inch rectangle. With a long side facing you, fold the dough in thirds and fit seam side down in the prepared loaf pan. Cover with plastic wrap and let rise in a warm place until doubled in bulk, about 2 1/2 hours.
Preheat the oven to 425° and set a rack in the center. In a small bowl, beat the remaining egg. Brush the top of the brioche with some of the egg wash and make a 1/4-inch-deep slit down the center of the loaf. Bake for 20 minutes. Brush the top again with egg wash and bake for 20 minutes longer, until the top is deep golden and an instant-read thermometer inserted in the center of the loaf registers 182°. Transfer the brioche to a rack to cool for 30 minutes, then unmold and let cool completely.