Bring milk, ¾ cup water, cubed butter, sugar, and salt to a simmer in heavy-bottomed saucepan, stirring frequently to avoid scorching the milk.
Transfer immediately to the metal bowl of a stand mixer, and cool to room temperature (30-45 minutes).
Once cooled, in a separate bowl, sprinkle the yeast over the remaining ¼ cup of (lukewarm) water (105-110 F). Let stand for five minutes. Stir to combine the yeast and water, then add to the milk mixture.
Add egg. Using the paddle attachment, beat on low to combine.
Gradually add the flour at low speed, scraping the bowl sides to incorporate all the flour.
Once a sticky dough forms, increase to high speed and beat for 30 seconds.
Replace the paddle with the dough hook attachment (or use your hands) to knead for 8-10 minutes.
Cover tightly with plastic wrap and refrigerate for 4 (minimum) to 8 (~max) hours.