To the bowl of a stand mixer fitted with the dough hook attachment, add milk, sugar, and yeast. Mix on low speed to combine.
With the mixer still on low, add eggs, mixing until combined.
Turn off the mixer. Add flour. Then, continue mixing at low speed. Once the flour is incorporated, with the mixer still on low speed, add butter, 1 piece at a time, blending well between each addition.
Once the butter is all incorporated, add the salt. Knead on medium speed until the dough that gathers on the hook is a soft smooth ball, about 5-6 minutes. At this point, you may need additional flour. I mixed in 1 tablespoon of flour at a time until the dough pulled away from the sides of the bowl, creating a smooth ball (a total of ¼ cup extra flour). The dough should be very soft and not sticky when handling it.
Spray a medium bowl with nonstick cooking spray. Place the dough into the bowl. Then, turn the dough over so that the top of it gets coated with cooking spray as well. Cover. Let rise in a warm area until it is doubled in size, about an hour.
With the cover still in place, place the bowl of dough in the refrigerator for 8 hours, up to overnight. This will slow the fermentation and chill the butter, making the dough easier to shape.
After 8 hours, remove the dough from the refrigerator and let it sit out at room temperature for 30 minutes.
When ready, place the dough on a lightly floured surface. Evenly divide it into 8 pieces. Shape each piece into a small ball.
Spray a 9x5-inch loaf pan with cooking spray.
Arrange the dough balls in a single layer, spacing them evenly to fill the loaf pan, with 2 rows of 4. Cover. Let it rise in a warm area for about 2-2 ½ hours, or until doubled in size.
Preheat the oven to 350°F.
In a small bowl, whisk the egg and milk together. Brush the top of the loaf with the egg and milk mixture.
Bake for 35-40 minutes, or until the internal temperature reaches 190°F. If the top is getting too dark, cover it with aluminum foil.
When done baking, let the brioche rest in the pan for 5-10 minutes. Then, remove from the pan and let cool completely on a wire rack.