Step 1Sprinkle the yeast over the warm milk in a mixing bowl. Stir until the yeast is completely dissolved.
Step 2Stir in the egg and 1 cup of the flour until blended. Sprinkle the remaining 1 cup of flour over the sponge to cover.
Step 3Set the sponge aside to rest 30 to 40 minutes. After resting, the flour will look crackly.
Step 1Add the sugar, salt, 4 beaten eggs and 1 cup of flour to the sponge. Using the dough hook on a stand mixer, mix on low speed for a minute or two, just until the ingredients look as if they're about to come together. Still mixing, sprinkle in one-half cup flour. When the flour is incorporated, increase the mixer speed to medium and beat for about 15 minutes, stopping to scrape down the hook and bowl as needed. During this mixing period, the dough should come together, wrap itself around the hook, and slap the sides of the bowl. If, after 7 to 10 minutes, you don't have a cohesive, slapping dough, add up to 3 tablespoons flour. Continue to beat, giving the dough a full 15 minutes in the mixer -- don't skimp on time; this is what will give the brioche its distinctive texture.
Step 2In order to incorporate the butter into the dough, you must work the butter until it is the same consistency as the dough. Smash the butter with a rolling pin or work it with a dough scraper until it is soft and pliable. When it's ready, the butter will be smooth and still cool, not warm, oily or greasy.
Step 3With the mixer on medium-low, add the butter a few tablespoons at a time. When all of the butter has been added, raise the mixer speed to medium-high for a minute, then reduce the speed to medium and beat the dough until you once again hear the dough slapping against the sides of the bowl, about 5 minutes. Clean the sides of the bowl frequently as you work. If it looks as though the dough is not coming together after 2 to 3 minutes, add up to 1 tablespoon more flour. When you are finished, the dough should still feel somewhat cool. It will be soft and still sticky and may cling slightly to the sides and bottom of the bowl.
Step 4For the first rising transfer the dough to a very large buttered bowl, cover tightly with plastic wrap, and let it rise at room temperature until doubled in bulk, about 2 hours.
Step 5For the second rising deflate the dough by punching it down in the bowl and folding it over. Transfer the dough to a lightly floured board and knead a few times. Divide the dough in half to make 2 (8-inch) brioche. (Keep the dough that you are not working with covered in the refrigerator.) Shape three-quarters of the dough from each half into a ball on a lightly floured surface and place into 2 (8-inch) buttered nonstick brioche pans. Roll each of the remaining balls of dough into a pear shape. Use your fingers to make an indentation in the center of one brioche and fit the narrow end of the ball into the depression. Repeat with the remaining dough.
Step 6Cover the dough lightly with buttered plastic wrap or an inverted bowl and refrigerate. Let rise until doubled in bulk, 4 to 6 hours or overnight. Remove from the refrigerator and let stand at room temperature. Let rise until doubled in bulk, about 1 hour.
Step 7Heat the oven to 375 degrees.
Step 8Brush the top of the brioche with beaten egg, taking care not to let the egg dribble onto the pan. Bake until golden brown, about 30 minutes. If the bread is browning too quickly, cover it lightly with foil. To test for doneness, insert a thermometer in the bottom of the bread. It should reach 200 degrees.
Step 9Let the brioche cool in the pans about 5 minutes, then remove the bread from the pans and let cool on a wire rack.
To make small brioche using 3-inch brioche molds, prepare the dough as directed. Set aside one-third of the dough and keep refrigerated. Roll the remaining dough into 24 golf ball-size rounds for the base of the brioche, and place one into each of 24 lightly buttered 3-inch nonstick brioche molds. Make an indentation in the center of each. Remove the leftover dough from the refrigerator and shape into 24 grape-size balls. Place a "grape" in each indentation. Cover the dough lightly with buttered plastic wrap and refrigerate. Let rise until doubled in bulk, 4 to 6 hours or overnight. Remove from the refrigerator and let stand at room temperature. Let rise until doubled in bulk, about 30 minutes, and brush with the beaten egg. To bake, place the small brioche molds on a baking sheet and bake at 375 degrees until golden brown, about 15 minutes. Allow to cool as directed above.