THE ONIONS
Start to cook the onions at least 1 day in advance: Cut off both the root and stem ends of each onion, peel, and leave whole. Put the onions and butter in a slow cooker, set the pot to high (or just turn it on if your pot doesn't have heat settings), and cook the onions for 24 hours, or until they are a deep golden brown. To get a really rich color, you may want to cook the onions even longer. (If you do not have a slow cooker, you can place the onions and butter in a roasting pan and bake in a 275°F oven, stirring occasionally, until they are deeply browned and caramelized.)
Turn off the pot, remove the cover, and allow the onions to cool in the liquid. Lift the onions out of the pot with a slotted spoon, transfer them to a colander set over a bowl, top with a plate, and weight the onions down-a can will do the trick here-to extract as much liquid as possible. Your goal is to drain the onions, not mash them, so take care. Let the onions drain for at least 1 hour, then remove them to a cutting board and, with a very sharp thin knife, cut them in half horizontally Save the liquid for flavoring soups or stews. The onions can be prepared up to a week ahead and kept covered in the refrigerator They can also be wrapped airtight and frozen for 3 months.
THE BRIOCHE
Start the brioche at least 24 or up to 36 hours ahead. Put all the ingredients for the dough in the bread machine, program the machine for "dough" or "manual," and press Start. Expect the dough to be sticky at the start but to become less sticky during kneading. If the dough, which will be very soft, does not form a ball after the first few minutes of kneading, add I tbsp more flour-no more. Some of the dough will stick to the bottom of the pan and all of the dough will seem very loose-this is just as it's supposed to be. The dough will become firmer when it is chilled.
Chilling the Dough
After the final knead, remove the dough from the machine and put it into a plastic bag; refrigerate for at least 24 or up to 36 hours.
At least 4 hours before you want to serve the Brie in Brioche, butter a 9-inch springform pan.
Assembling the Pastry
Turn the cold brioche dough out onto a lightly floured work surface. Cut off one third of the dough and roll it into a 12-inch circle about 1/16 inch thick. Fit the dough into the bottom of the prepared pan, push-ing the dough out from the center so that it covers the bottom and goes up the sides of the pan a ittle. Brush the sides of the dough with a little of the egg wash.
Center the Brie on the dough. Arrange the onions, cut side down, on the Brie. If you have some onions left over, you can either reserve them for another use or make a second layer of onions over the Brie.
Cut off one third of the remaining dough and roll it into a 14-inch circle. Center the dough over the onion-covered Brie and work your way around the dough, tucking the sides in to meet the bottom piece of dough. The top and sides will probably be bumpy-don't worry about them; they'll even out as the dough rises.
Roll the remaining piece of dough into a strip about 30 inches long and 4 to 6 inches wide. Make 3 cuts the long way, leaving the lengths attached at the top. Starting at the attached end, braid the strips together, then position the braid around the edge of the brioche, like a wreath.
Rise
Brush the dough with the egg wash and place the pan, uncovered, to rise at room temperature until doubled in volume, about 40 minutes.
While the dough is rising, position a rack in the lower third of the oven and preheat the oven to 425°F.
Baking the Pastry
Bake the brioche for 15 minutes, then lower the oven temperature to 375°F and bake for 30 minutes more. If, as sometimes happens with very buttery breads, the top starts to turn very brown before the baking time is completed, cover it loosely with a foil tent.
Cool the brioche for at least 30 minutes before serving, or cool it to room temperature and serve.
Storing
The dish is best served the day it is made. Leftovers can be covered and stored overnight in the refrigerator and brought to room temperature before serving.