In a small saucepan over medium heat, warm the milk to approximately 110℉. Use a kitchen thermometer to check the temperature, being sure not to let it boil.
⅔ cup whole milk
In the bowl of a stand mixer, combine 2 ½ cups of the flour, ¼ cup sugar, yeast, salt, warm milk, ¼ butter, and one egg. Slowly stir to combine.
3 cups all-purpose flour, ½ cup sugar, 1 package active dry yeast, 1 teaspoon salt, ⅔ cup whole milk, 2 egg, 1¼ cup unsalted butter
Insert the dough hook attachment to your stand mixer, and set to a medium or medium-high speed to knead the dough for about 10 minutes. When the dough is ready, it should pass the windowpane test. (See notes.)
Spray a large bowl with oil or Pam cooking spray, and put the dough into the bowl. Cover with plastic wrap and put it in a warm place. Allow the dough to rise for about 90 minutes, or until doubled in size.
Punch down the dough, then turn it out onto a clean surface and divide into eight equal parts. Roll each part into a ball, the flatten with your hands or the bottom of a large glass. You can use a bit of flour if you have trouble with the dough sticking to your hands.
Cut the remaining 8 tablespoons of butter into 8 equal pieces, then place one on the top of the dough. Use your fingers to wrap the dough around the butter. (If needed, you can cut each piece of butter into smaller pieces!)
1¼ cup unsalted butter
Place the dough seam side down on top of a silicone mat or parchment paper on a baking sheet. Cover with buns a tea towel, and allow them to rise again for about 45 minutes.