¼ cup warm (110º) whole milk
¼ cup warm (110º) water
3 tablespoons sugar
4 teaspoons active dry yeast
2¾ cups all-purpose flour
1½ teaspoons salt
3 large eggs, at room temperature, lightly beaten
12 tablespoons (1½ sticks) butter, at room temperature
1 large egg lightly beaten with 1 teaspoon cold water, for glaze
This dough does have to sit overnight, so make sure to leave time for that when you make it.
Combine the milk and water in the bowl of a stand mixer fitted with the dough hook. Add a pinch of sugar, then sprinkle over the yeast. Allow to sit until yeast is dissolved.
In a small bowl, whisk flour and salt together and set aside. Using a rubber spatula, stir the yeast until the mixture looks creamy. Add the flour mixture all at once, and turn the mixer on low to dampen the flour. Increase speed to medium-low and mix for 1 to 2 minutes, scraping the sides and bottom of the bowl as needed. The dough will be dry and messy at this point.
Turn mixer to low and add the beaten eggs one at a time, mixing until each egg is incorporated before adding another. Mix in the remaining sugar, increase speed to medium and continue beating for about 3 minutes, or until the dough starts to come together. Reduce speed to low and add butter in 2-tablespoon pieces, beating for about 30 seconds before adding more. Once all the butter is incorporated, the dough will be very soft. Increase speed to medium-high and beat for 8-10 minutes, until the dough pulls away from the sides of the bowl and climbs the dough hook.
Transfer the dough to a large, lightly buttered bowl. Cover with plastic wrap and let dough rise until nearly doubled in size, at least an hour. The length of time this takes will depend on the warmth of your room. To deflate the dough, gently lift the edges and allow it to fall back in. Cover the dough with plastic wrap and refrigerate, deflating the dough every 30 minutes for 2 hours. Press plastic against the surface of the dough and refrigerate overnight.
When ready to make rolls, butter 12 muffin cups. Divide the dough into 12 equal pieces (a scale works wonders here). Cut each piece into thirds. Roll each piece on a lightly floured work surface until a smooth ball forms. Place 3 balls into each muffin tin. Cover with wax paper and allow to rise in a warm place for 1 to 2 hours, until almost doubled. The dough will rise above the muffin cups.
Heat the oven to 400º. Gently brush the tops of the rolls with the egg wash, taking care not to let it drip between the dough and the sides of the muffin cups. Place on a baking sheet and bake for 20-23 minutes, covering brioches with a foil tent if they start browning too quickly. Transfer pan to cooling rack and allow to cool for 10 minutes before removing brioches. Serve warm or at room temperature.