Combine 2 cups of the flour, the yeast, water and eggs in the bowl of a stand mixer. Beat to combine, then beat on high for 1 minute, until smooth. Cover and let rest for 45 minutes.

Add the remaining 2 cups of flour, the Parmesan, garlic, sugar and salt. Beat on low to incorporate. Cut the stick of butter in half lengthwise, then slice into 1/2 inch slices. Put the mixer on medium and beat in the butter, a few bits at a time, until it is fully incorporated. Cover the bowl and place in the refrigerator for at least 4 hours or overnight.

Divide the dough into 8-10 equal pieces. Shape into balls. Place the balls on greased or parchment paper lined baking sheets, 4 to 5 buns per sheet. Flatten the balls with the palm of your hand to about 3/4” high Cover and let rise until they are doubled in size, 30-45 minutes.

Preheat the oven to 350ºF.

Mix the egg yolk and the water together. Brush over the tops of the buns, then sprinkle with additional Parmesan, if desired.

Bake in the preheated oven for 20-25 minutes, or until browned. Remove from the pan and allow to cool completely on a wire rack.