Combine milk, sugar, yeast, and 1/2 cup bread flour in the bowl of a mixer. Mix until just combined. Sprinkle with remaining 3/4 cup plus 2 tablespoons bread flour, the all-purpose flour, and salt to cover; do not mix. Let stand for 45 minutes.
Using the dough-hook attachment, mix dough to combine. Add 4 eggs, and continue to mix until dough is smooth and does not stick to sides of bowl, about 5 minutes. Mix in remaining egg. Add butter, 1 tablespoon at a time, mixing until incorporated. Continue mixing until dough is smooth and comes together in a ball around the dough hook, about 5 minutes more.
Coat a large bowl with cooking spray. Transfer dough to bowl, and cover with plastic. Let stand in a warm place until doubled in volume, about 1 1/2 hours.
Punch down dough, and re-cover. Refrigerate for 1 1/2 hours. Punch down dough again, re-cover, and refrigerate overnight.
Preheat oven to 425 degrees. Turn out dough onto a lightly floured surface, and punch down. Working in batches (refrigerate remaining dough as you work), evenly divide dough into thirty-two 1-ounce pieces. Punch down each piece using the palm of your hand, and press each into a tight ball using the heel of your hand.
Butter small Turk's-head or similar shaped cast-iron pans. Place dough balls inside molds (dough should fill three-quarters of each mold section). Loosely cover pans with plastic; let stand in a warm place until dough balls are soft and springy to the touch, about 30 minutes.
Lightly brush tops with egg wash. Bake for 10 minutes. Reduce heat to 350 degrees, and continue to bake until tops are dark gold (time will vary depending on size of pans). Remove from oven, and immediately remove brioche rolls from pans, by gently tapping, and transfer to a wire rack. Let cool completely.
Freeze for up to a month.
On the Martha Stewart Show, Martha used this recipe to make 12 larger buns. In step 6, butter twelve 3-ounce brioche molds and place on a large baking sheet; place 3 dough balls inside each mold.