Proof the yeast by combining the yeast, milk, and sugar in a small bowl. Mix together to dissolve the sugar.
Add the 110 F water and let the mixture sit for 5 minutes at which point there should be a creamy foam on the surface of the liquid and all the yeast should be dissolved.
Using the paddle attachment, beat the butter on medium speed in a stand mixer until light and fluffy
Slowly mix in the dry ingredients. The mixture should resemble crumbs when you are finished.
In a separate bowl, beat the 5 eggs with a fork
VERY slowly, add the eggs to the crumb mixture in the stand mixer-low-medium speed. The dough should form a ball around the paddle in the stand mixer.
Add ¼ cup of flour to the dough if too wet.
Invert the bowl onto a lightly floured surface.
Form the dough into a ball and place into a large, greased bowl.
Cover the dough with a damp towel and let rise for 2-3 hours.
Once the dough has doubled in size, punch the dough down.
With floured hands, invert the bowl onto a lightly floured surface.
Cut into 8 equal pieces.
Form the buns by folding the sides of the buns inward.
Place the buns onto a silpat or parchment paper-lined cookie sheet or tray.
Cover the buns and let them rise for 1½ hours.
Pre-heat the oven to 400 F
Apply an egg wash using 1 egg and a sprinkling of water. Sprinkle the buns with sesame seeds.
Place into the oven for 15-20 minutes, watching the buns so they dont burn.
Remove from the oven and serve!
Recipe by THE GRUMPY GOURMAND at https://www.thegrumpygourmand.com/2015/01/23/homemade-brioche-buns/