Hungarian coffee cake became known as monkey bread. It is a beloved dessert and breakfast treat!
Proof the yeast: in a 1 cup Pyrex measuring cup, add the water, yeast, and sugar; set aside to proof for 5 - 10 minutes.
In the bowl of an electric mixer fitted with a paddle attachment, add the proofed yeast, cooled milk, shortening, sugar, salt, eggs and flour; blend together. Switch the paddle attachment to the dough-hook attachment, knead the dough until smooth, possibly adding a bit of flour. At this point, do not leave the Kitchen Aid mixer. It can rock off the table!
Add a couple of teaspoons of corn oil to a bowl. Put the dough in the bowl, turning to coat with the oil. Cover with a clean dish towel; set aside in a warm, draft-free place for 1 1/2 - 2 hours. Punch down dough. Set aside for a second rise; 1 1/2 - 2 hours.
On a lightly floured silicone baking mat, pour out the dough. Using a pastry scraper, cut the dough into walnut-size pieces, brush with butter, dip each piece in the sugar/cinnamon mixture, and drop into an Angle Food cake pan. Cover with a clean dish towel, set aside in a warm, draft-free place for 30 - 45 minutes to rise.
During the rising, preheat oven to 375ºF.
Bake for 30 - 40 minutes, watching carefully towards the end. If it begins to brown too much, cover lightly with a piece of aluminum foil.
When the coffee cake is fully baked, the internal temperature should be between 195ºF. - 205ºF.
Cool on a rack.
Remove from pan, smooth the frosting over the top and just over the top's edges.