Make the Dough: In a medium bowl, whisk together the melted butter and 4 tablespoons of the sugar until the sugar has dissolved. Then, whisk in the salt and sour cream until smooth. Set aside.
Activate the Yeast: In a mixing bowl, add ¼ cup warm water, instant yeast, and the remaining 1 tablespoon of the sugar; stir. Allow the yeast to sit for about five minutes until it bubbles.
Combine Ingredients: Whisk the warm butter and sour cream mixture into the yeast mixture, along with the egg and 1 cup of the flour. Whisk until smooth. Using an oiled wooden spoon, begin mixing in small amounts of the remaining flour until you form a soft dough. This will take about another 2 cups of flour. You want the dough to be tacky, but not sticky.
Knead the Dough: Turn the dough out onto a clean surface lightly dusted with flour. Knead the dough until smooth and elastic, about 5 to 10 minutes, adding more flour by the teaspoon if needed.
First Rise: Place the ball of dough into a large, well-oiled bowl, then flip the dough so all of the surface area of the dough is oiled. Cover the bowl with oiled plastic wrap and a hand towel, then set the bowl in a warm, draft-free area and allow the dough to rise until it has doubled in size, about 1 hour.
Make the Filling: While the dough is rising, make the filling. Combine the melted butter, cinnamon, and granulated sugar in a medium bowl and stir to fully combine.
Roll Out and Fill the Dough: Once the dough has doubled in size, turn it out onto a lightly floured surface. Lightly flour the dough and a rolling pin. Roll the dough into a rectangle about 18 inches long by 14 inches wide. Spread the cinnamon sugar mixture evenly over the dough, leaving a 1-inch border around the outside of the dough.
Shape the Cake: Roll the cake up jellyroll-style and pinch the seams shut. Carefully move the roll to a parchment-lined baking sheet, seam-side down. Bring the ends together to form an oval and press the edges together to completely seal the cake into an oval.
Second Rise: Once again, cover the cake with oiled plastic wrap and a hand towel and allow it to rise for another 30 minutes.
Bake the Cake: While the cake is rising, preheat the oven to 375 degrees F. Once 30 minutes have passed, remove the plastic wrap and hand towel from the cake and bake in the upper third of the oven until the cake is golden brown, about 25 minutes. Immediately transfer the cake (with the parchment paper underneath) to a cooling rack after removing it from the oven. Allow the cake to cool for at least 20 minutes before icing the cake.
Make the Icing: Once the cake has cooled for 15 minutes, make the icing. Whisk together the powdered sugar, whole milk, salt, and vanilla, until smooth. You want the icing to be able to drizzle easily but not just run right off the cake, so if the icing is too thin, whisk in more powdered sugar and if the icing is too thick, whisk in a touch more milk.
Decorate the Cake: Move the cake to whatever platter you wish to serve it on. (At this point, stick a dried bean or little plastic baby into the cake through the bottom. It's tradition in Louisiana that whoever gets the baby has to spring for the next cake! Elsewhere, it's a sign of good luck.) Slide pieces of wax paper under the cake so that it can catch any icing or sugar drips. Drizzle the icing evenly over the cake and allow it ooze down the sides. Before the icing has a chance to set, sprinkle on rotating strips of colored sugar.
