Heat butter on medium, stirring occasionally. The butter will bubble and foam until it reaches a light brown color and has little browned bits on the bottom.
Mix the buttermilk and sugar in a mixing bowl then sprinkle yeast on top.In a separate bowl mix the brown butter, egg, egg yolk and salt then whisk into the yeast mixture.
Place bowl on a stand mixer fitted with a dough hook then add about 1/2 cup of flour. Mix on low speed until mostly combined then repeat until all the flour is gone.
Knead dough for an additional 5 minutes. Transfer to a large greased bowl then cover and allow to rise in a warm place for an hour or until doubled in size.
Tightly roll the dough towards you then cut 12 even rolls and place in a lightly greased 9×13 pan. Cover and allow to rise in a warm place for an hour or until doubled in size.