1. Place 1 1/2 cups (6 1/4 ounces) of the flour, the yeast, water, and eggs into the bowl of a mixer or the bucket of your bread machine. Beat at medium speed (or knead in the bread machine) until smooth. Cover the mixture and let it sit for 45 minutes.
2. After 45 minutes, the sponge will have developed some bubbles, but not risen much because the mixture is thin. The yeast is getting a jump start.
3. Add the remaining 1 1/4 cups (5 1/4 ounces) flour; the sugar, and salt.
4. Beat for 8 to 10 minutes (switch to a dough hook if you're using a mixer), or knead in the bread machine, until the dough cleans the sides of the bowl and becomes shiny and elastic.
5. With the mixer or bread machine running, add the soft butter 2 tablespoons at a time, letting the butter become absorbed before adding the next chunk. Repeat until all of the butter is added. Cover the dough and let it rise for 1 hour. It'll be very soft at this point, and should have grown by about a third. Turn the dough out onto a lightly floured surface and gently fold it over several times (use a bench knife to scrape up any bits that stick to the table). Place the dough in a greased bowl, cover the bowl, and refrigerate it for a minimum of 4 hours and up to about 16 hours. The dough will firm up considerably.
6. To make large brioche, remove the dough from the refrigerator, take a piece of the dough the size of a golf ball and set it aside. Form the rest into a round loaf. Grease a brioche pan and place the loaf into it. Take a small bowl and grease the inside and the outside of it; place this little bowl on top of the dough, and shape the reserved small piece of dough into a ball. Put this ball inside the bowl and allow the dough to rise for 1 to 1 1/2 hours, until it's an inch above the edge of the pan and looks puffy. Preheat the oven to 375°F.
7. Once the dough is risen, remove the small bowl and gently replace the topknot. Whisk together the egg and water, and brush the bread with the egg glaze. Bake for 40 minutes, until the center of the bread tests 190°F when measured with an instant-read thermometer. Check the bread after 20 minutes: if it's browning too quickly, tent the top with aluminum foil. Traditional brioche is a very deep golden brown, as the photo illustrates, but if you want a lighter crust the foil tent will do the trick. Remove the bread from the oven and cool on a rack for 10 minutes before loosening the edge of the bread from the pan and tipping it out. Place the bread on a rack to finish cooling completely before slicing.
8. To make individual brioche, divide the dough into 12 pieces. Take a marble-sized piece of dough from each of the 12 pieces and set it aside. Roll each larger bit of dough into a ball; place them in greased brioche tins or paper pans. Roll the small pieces into balls. Set them aside and let them rise with the small brioche for 45 minutes to 1 hour.
9. When the dough has risen and is puffy-looking, preheat the oven to 375°F. Grease your finger, and poke a "belly button" into the center of each individual brioche. Gently place the small marble of doughs into these dents, then brush the tops with egg wash. Bake the brioche for 25 to 30 minutes, until the centers test 190°F when measured with an instant-read thermometer. Remove the brioche from the oven an cool on a rack before serving.