Put the yeast, water, and milk in the bowl of a stand mixer and, using a wooden spoon, stir until the yeast is dissolved. Add the flour and salt, and fit the mixer with the dough hook, if you have one. Toss a kitchen towel over the mixer, covering the bowl as completely as you can to keep you from being covered in flour! Turn the mixer on and off in a few short pulses, just to dampen the flour (you can peek), then remove the towel, increase mixer speed to medium-low and mix for a minute or two. At this point you will have a dry, shaggy mess.
Scrape the sides and bottom of the bowl with a rubber spatula, set the mixer to low and add the eggs, followed by the sugar. Increase the mixer speed to medium and beat for about 3 minutes, until the dough forms a ball. Reduce speed to low and add the butter in 2 Tablespoon size chunks, beating until each piece is almost incorporated before adding the next. You’ll have a dough that is very soft, almost like a batter. Increase the speed to medium-high and continue to beat until the dough pulls away from the sides of the bowl, about 10 minutes.
Transfer the dough to a clean bowl, cover with plastic wrap and leave at room temperature until nearly doubled in size, 40-60 minutes. Deflate the dough by lifting it up around the edges and letting it fall with a slap into the bowl. Cover the bowl with plastic wrap and put it in the refrigerator. Slap the dough down in the bowl every 30 minutes until it stops rising, about 2 hours, then leave the covered dough in the refrigerator overnight.
If making Brioche loaves: Butter and flour 2 8.5 X 4.5 inch loaf pans. Pull dough from the fridge and divide into two equal pieces. Cut each piece of dough into 4 equal pieces and roll each piece into a log about 3.5 inches long. Arrange 4 logs crosswise in the bottom of each pan. Put the pans on a baking sheet lined with parchment or silicone mat, cover the pans lightly with wax paper and leave the loaves at room temperature until dough fills pans.
Preheat oven to 400 degrees F. Bake until loaves are well risen and deeply golden, about 30-35 minutes. Cool for 15 minutes, then run a knife around the sides of the pans and turn the loaves out onto cooling racks. Cool at least one hour.