In a small bowl, stir 2 1/4 teaspoons instant yeast and 1 teaspoon granulated sugar into 1/2 cup warm milk. Let stand 5–10 minutes, until foamy.
In the bowl of a stand mixer (or a large bowl), whisk together 4 cups bread flour, 1/3 cup granulated sugar, and 1 1/2 teaspoons salt. Form a well in the center.
Add the yeast mixture and 3 large eggs to the well. Using the dough hook attachment (or a wooden spoon), mix on low speed until the dough comes together. Increase to medium speed and knead for 10–15 minutes, until the dough is smooth and elastic.
With the mixer running on medium speed, add 2/3 cup unsalted butter, one cube at a time, kneading until each piece is fully incorporated. Continue kneading until the dough is soft, shiny, and passes the windowpane test, about another 10–15 minutes.
Shape the dough into a ball and place it in a lightly greased bowl. Cover and let rise in a warm spot until doubled in size, about 1 1/2–2 hours.
Punch down the dough, cover, and refrigerate for at least 2 hours or overnight.
Divide the dough into equal pieces and braid them or place in a greased loaf pan. Cover and let rise until almost doubled, about 1–1 1/2 hours.
Preheat the oven to 350°F (180°C). Beat the egg reserved for the egg wash.
Brush the top of the loaf with the beaten egg. Bake for 25–35 minutes, until golden brown and hollow-sounding when tapped on the bottom.
Remove from the pan and cool completely on a wire rack before slicing.