Pour the milk and 311g/10.97oz of the eggs into a 5.76-L/6-qt mixer bowl fitted with a dough hook and stir to combine. Pour the bread flour, salt, sugar, and yeast on top. Mix on low speed until just incorporated.
Add 173g/6.1oz of the butter and switch the mixer to medium speed. Once that butter has been incorporated, add another 173g/6.1oz of butter. Wait until it has been completely mixed in, and then add the remaining butter.
Continue to mix on medium speed until full gluten development is achieved. To check for gluten development, perform a “window test” (see page 93); stretch a small amount of dough with your hands. It should be elastic enough to be pulled until it is very thin and you can see through it without it ripping.
Add the remaining eggs and mix until just incorporated. At this point, the final dough temperature should not exceed 27°C/80°F.
Remove the dough from the bowl and place it on a floured surface. Cover with plastic wrap and let it bulk ferment for 45 minutes.
Transfer the dough to a sheet pan lined with silicone paper or greased parchment paper to prevent it from sticking. Wrap the sheet pan with plastic wrap and refrigerate. The dough is ready to be shaped when it is has firmed up and relaxed for at least 1 hour. At this point, it can be reserved in the refrigerator for 12 hours before shaping, or it can be frozen for later use.
Coat the interior of 3 half-cylinder terrine molds measuring 5cm/2in wide by 40cm/16in long by 4cm/1.5in deep with a light coat of nonstick oil spray, and then line them with silicone paper.
Divide the dough into three 300-g/10.6-oz pieces. This is all that is needed for 10 servings; the remaining dough can be frozen. Making smaller amounts of brioche is not only inefficient, but you will also not be able to obtain a properly mixed dough. It is a dough that keeps very well in the freezer if it is well wrapped.
Flatten the piece of brioche with the heel of your hand, using some flour on the work surface and on the brioche to keep it from sticking. Using a rolling pin, roll it out to 10cm/4in wide by 37.5cm/15in long. Place the dough in front of you, with the long side horizontally placed. Roll the dough toward you, packing it tightly with your fingertips every time it comes full circle. When you reach the seam, seal it tight by pinching it sealed to the body of the dough. Flatten the dough again, making sure the seam is facing up.
Transfer the dough to the prepared mold with the seam facing down. Flatten the tube of dough.
Combine all of the ingredients for the egg wash in a small bowl using a whisk, then pass the mixture through a fine-mesh sieve. Brush the dough with the egg wash.
Proof the dough for 1 hour and 45 minutes, or until doubled in size, in a 32°C/89°F environment. The dough can also be proofed for 3 hours at room temperature (21°C/70°F); if using this method, keep the dough covered with a plastic bag, making sure the plastic bag does not touch the surface of the dough.
Preheat a convection oven to 160°C/325°F.
Once the dough is proofed, brush it again with the egg wash and, using a sharp pair of scissors, snip small incisions in a straight row across the crown (the highest point of the middle section of the dough) of the dough.
Bake until dark golden brown, about 13 minutes.
Remove the dough from the oven and quickly extract it from the mold. It will collapse onto itself if it is left to cool in the mold.
Once it has cooled off completely, it can be used for service.