Cube your butter into chunks and make sure it is soft to the touch before you proceed.
Combine all of your ingredients into a big bowl, except for half of the butter and the salt. I like to crack my eggs in a separate bowl, and then add the water to them before adding to the rest of the ingredients.
Use your hands to squeeze everything together. It's going to be messy and sticky, so make sure your bowl is big enough to where you aren't making a mess.
If you absolutely feel you need to add more flour, now is the time to do it. I would add small amounts at a time, but don't go overboard. Your hand kneading or machine mixing process, with time, will give you the necessary gluten development! Trust the process.
If you are machine mixing, start at the lowest speed and mix for 2 minutes. Crank the machine up to medium speed and mix for 2-3 minutes or until you see the dough start to come together. Let the dough rest for 1 minute, and then add the rest of your butter before switching it to the highest speed setting and let it mix until the dough slaps on the side of the bowl and has a smooth surface. This can take 10-15 minutes.
If you are hand mixing, first add only half of the butter and squeeze until all ingredients are incorporated. Once you have a consistent texture, you can begin to stretch and slap the dough on the table, while slowly incorporating the rest of the butter. It will come apart very easily, but the longer you knead/stretch/slap the dough, the more it will come together. You can give it a few periods of rest (5 or so minutes) to let it come together.
Remember: with a machine mix, it takes 15 or so minutes for the dough to come together. If you opt to hand mix brioche, be prepared to knead and develop your dough for at least 30 minutes. Flour your work surface as well, to ensure that your dough doesn't stick too much.
Once your dough is fairly smooth on the surface and has come together, place it in a clean and oiled bowl for bulk fermentation. If you hand-mixed, you can give it a couple of stretches and folds every 30 minutes to continue to help the gluten formation process.
Allow your dough to go through the initial rise in a warm place for 6 hours. My house temperature is usually between 72-75f.
After you notice your dough is smooth and springy to the touch, put it into the fridge overnight to continue developing flavor and a strong structure. I usually will refrigerate it for about 10 hours.
Once you complete the cold stage of your fermentation, remove the dough. Now it's time to get creative! You can divide and shape into whatever shape you want. For buns, I usually divide into 130-gram piece and round them into little balls. You can put them side by side on a baking tray in a 4 x 6 pattern. It's ok for them to proof into each other because they will bake nicely and pull apart. For pull-apart bread in a loaf tin, use 6 110 gram pieces and squeeze together into the pan in two rows of 3. For a braided loaf, divide into your desired size strips and roll them out so that you can braid your loaf.If you want to top your brioche with savory or sweet things, now would be the time to do it. I've used cinnamon sugar, tomatoes, ham and cheese, and pretty much anything I could think of. Be creative here, but remember that plain old brioche is probably the best.
Brush your dough with an egg wash and proof for about 4 hours, or until the dough grows noticeably in size and becomes springy to the touch. You'll also start to get a great aroma from the dough.
Preheat your oven to 375f, and egg wash your brioche one final time.
Bake your brioche for 20-25 minutes. Depending on your oven and color preference you may need to go a bit longer. If you're using convection, 20 minutes will usually do.
Enjoy your warm sourdough brioche! If you plan to fill your brioche rolls after baking, allow them to cool completely before piping in your filling or cream.
