Combine the warm water, yeast and sugar in a 5-quart mixer. Let yeast become activated and foam, then add the milk solids, salt and 9 1/3 cups flour. Mix with dough hook until flour is just incorporated. Avoid overmixing as that causes rapid toughening of dough. Place dough mixture in airtight container with room for rising, and refrigerate at 38 to 40 degrees for 12 to 24 hours, punching down occasionally if dough rises too much.
Rolling butter into dough: Place 1 pound unsalted (or lightly salted) butter into 5-quart mixer. Mix with paddle dough hook, gradually adding remaining 1/2 cup flour. Mix until butter is moderately soft but not creamy in texture. Too soft butter will not layer properly in dough.
Remove dough from refrigerator and turn out onto well-floured surface. Spread dough with hands into a 6-by-8-inch rectangle. Shape soft butter into 3-by-4 inch rectangle. Place butter in middle of dough. Envelope butter with dough, bringing dough from sides into middle without overlapping; then dough from top and bottom into middle, again without overlapping. Press envelope of dough down evenly with hands, preserving rectangular shape.
Set aside to rest 15 to 20 minutes. (You may want to refrigerate the dough during the first rest period if butter is very soft.).
Turn envelope of dough and butter onto its "tummy " with seams down. Using a large rolling pin, roll rectangle of dough down to 3/8 to 1/2-inch thickness uniformly. Fold in thirds.
Turn the dough 90 degrees and place seam down on your rolling surface to rest 15 to 20 minutes more. Finally roll dough down again to 3/8 to 1/2-inch thickness. Fold in thirds. Place in a large plastic bag, carefully preserving its folded shape. Refrigerate 12 to 14 hours, again at 38 to 40 degrees.
Roll croissant dough into rectangle 12 inches wide and 1/8-inch thick. Relax by lifting with hands and let it contract on table surface. Length of dough determines the number of morning buns ultimately cut.
Wet exposed surface lightly with mixture of egg and water (proportions are roughly 1 egg per 1 quart water).
Spread brown sugar and cinnamon mixture (in proportions of 2 1/2 teaspoons cinnamon to 1 pound brown sugar) over entire surface of dough.
Note: Too much moisture from either water or melting brown sugar can overwhelm the dough during the baking process. Water mixture is only to help sugar and cinnamon adhere to dough. The butter in the dough will melt into the sugar.
Crimp long edge of dough closest to you as you begin to roll this dough up like a jelly roll into a tube. After having rolled your tube of dough, cut off slices 2 inches wide or to stand above your well-greased large muffin pans by 1/4 - 1/2 inch when placing them in cut side down.
Bake immediately or refrigerate overnight before baking. Bake in a 350-degree oven for 35 to 50 minutes or until puffed and dark brown. Check for doneness in center of buns. They should spring back. (It is possible to invert buns onto flat tray and finish the last few minutes of baking upside down.).
Let buns cool in pan a few minutes. Garnish buns by rolling them in white sugar and cinnamon in proportions of 1 pound granulated sugar to 2 1/4 tablespoons of cinnamon.
Serve warm, within 4 hours, or freeze immediately to serve warm later. Makes 24 Morning Buns.