1. To Make the Dough: Preheat the oven to 375 degrees F. In a small or medium saucepan over medium heat, add the butter, 4 tablespoons of the sugar and the salt; stir. Once the butter has melted, add the sour cream and heat to luke warm, about 105 degrees. Meanwhile, in a mixing bowl, add ¼ cup warm water, yeast, and 1 tablespoon of the sugar; stir. Allow the yeast to sit for about five minutes until it bubbles and becomes active.
2. Once the yeast is active, whisk in the warm butter and sour cream mixture, the egg, and 1 cup of the flour. Whisk until smooth. Using an oiled wooden spoon, being 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.
3. Turn the dough out onto a clean surface lightly dusted with flour. With oiled hands, knead the dough until and elastic, about 5 to 10 minutes, adding more flour by the teaspoon if needed.
4. 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.
5. While the dough is rising make the filling. Combine the melted butter, cinnamon and sugar in a medium bowl and stir to fully combine.
6. 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.
7. Roll the cake up jellyroll-style and pinch the seams shut.
Carefully move the roll to a parchment-line 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.
8. Once again, cover the cake with oiled plastic wrap and a hand towel and allow it to rise for another 30 minutes.
9. During the second rise, make the colored sugars. Place ½ cup of superfine sugar into three separate bowls. Using the food coloring, make one bowl of green sugar, one bowl of yellow sugar, and one bowl of purple sugar (by mixing the blue and red). Use the back of a spoon or a pestle, work the food coloring into the sugar by grinding it against the side of the bowl and working the coloring throughout all of the sugar. Continue this until the sugar is uniform in color and there are no clumps. I used 12 drops of food coloring total for each of the colors.
10. Once 30 minutes have passed, remove the plastic wrap and hand towel from the cake and bake in the upper third of the oven for about 25 minutes, or until the cake is golden brown. Immediately transfer the cake to a cooling rack after removing it from the oven. Allow the cake to cool for at least 20 minutes before icing the cake.
11. Once the cake has cooled for 15 minutes, make the icing. Whisk together the powdered sugar, salt, vanilla, melted butter and milk 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 to thin, just whisk in more powdered sugar and if the icing is too thick whisk in a touch more milk. After the cake has had a chance to cool, 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.
12. 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. Slide the wax paper pieces out from under the cake and discard. King cake can be served warm or at room temperature.