Instructions

1. In a mixer bowl combine the starter, water, milk and 2 cups of the flour. Mix to form a thick batter. Cover the bowl and set aside to rise for 30 minutes.
2. Add the sugar, melted butter and egg then mix to combine. Change to the dough hook if using a stand mixer. With mixer on low add the remaining flour, 1/2 cup at a time. The dough should be soft and elastic, will clear the sides of the bowl and gather on the hook. Knead for 2-3 minutes. If mixing by hand continue mixing and then kneading in the flour by hand. Dump the dough onto a lightly floured surface. Knead into a smooth ball.
3. Transfer the dough to a lightly oiled bowl, turning once to coat the dough. Cover the bowl and set it aside at room temperature. After 30 minutes uncover the bowl, lift one side of the dough over into the middle of the dough. Repeat with the other three sides of the dough then flip the dough over. Cover the bowl and after 30 minutes repeat the procedure. Cover the bowl and after 60 minutes repeat the procedure. Cover the bowl and after 60 minutes repeat the procedure one last time (total ferment time is 3 hours).
4. Generously brush a 9” x 13” baking pan with some of the melted butter, reserve the rest. In a small bowl, combine the granulated sugar, brown sugar and cinnamon. Turn the dough onto a floured surface without kneading out the air. Roll the dough to a 18" x 16" rectangle. Generously brush the surface of the dough with egg wash. You should use almost 1/2 the egg wash to get a good coating so the cinnamon sugar will stick. Sprinkle the cinnamon sugar over the egg washed dough. Use your hands to smooth out the cinnamon sugar to an even layer all the way to the edge of the dough.
5. Working from the long side, tightly roll the dough into a log, pinch the seam to seal. Straighten out the log so it's of even thickness and is 18" long (see note 1).
6. Use a serrated knife to cut the log into twelve 1.5" wide buns. Place the buns in the prepared pan. Generously brush the tops of the buns with the melted butter. Cover the buns with plastic wrap and refrigerate overnight (see note 2).
7. In the morning take the pan out of the refrigerator. You'll see some melted sugar in the bottom of the pan, this will form a sticky bottom as the buns bake. Allow the buns to proof at room temperature until they’re puffy and almost filling the pan, about 1.5 - 2 hours. The exact time needed for proofing will vary depending how active your starter was and the temperature of your kitchen. (see note 3)
8.  Preheat the oven to 375°F. Bake the buns until they're until golden brown and set in the center, 25-30 minutes. 
9. While the buns are baking, make the icing. Combine the butter and sugar in a mixing bowl and cream until well combined. Add the cream cheese and whip until well aerated. Add the vanilla and optional lemon juice. The lemon juice will spark the flavor of the icing but it's ok to go without.
10. Remove the buns from the pan to a serving plate while they are still warm. Spread cream cheese icing generously over the buns and indulge.