Mix 4 cups of flour, sugar, yeast, and salt together in the bowl of a stand mixer fitted with a dough hook.

In a medium bowl, whisk 3 eggs until combined.

Melt butter in a small saucepan over medium-low heat. Turn heat to low and add buttermilk to melted butter; heat until lukewarm (will be just warm to your touch). Whisk buttermilk mixture into beaten eggs.

Beat egg mixture into flour mixture on medium-low speed to make a firm, elastic dough, about 10 minutes. If dough doesn't separate from the sides of the mixing bowl during kneading, beat in ¼ cup more flour.

Turn dough out onto a work surface and knead for 1 minute, using more flour if needed to prevent sticking; form into a ball and place into a bowl coated lightly with oil or cooking spray. Turn dough around to lightly coat. Cover bowl with plastic wrap and let rise in a warm place until doubled, about 2 hours.

in a medium bowl, combine the almond flour, sugar, salt, and orange zest. Mash in the butter until it’s completely incorporated. Stir in the eggs one at a time, along with the rum and almond extract. Don't be concerned if the mixture is not completely smooth. Cover and chill.

Preheat oven to 350 degrees.

Punch down dough and turn out onto a lightly floured piece of parchment paper. Roll dough into a 10x28-inch rectangle. Spread almond filling over the rectangle, leaving a 1-inch border along each edge - use a thin layer of filling otherwise it will seep out when trying to roll it and seep out more when baking. Lift an edge of the parchment paper sheet to roll dough into a log shape. Pinch seams closed to seal in filling, keeping roll on the parchment paper.

Form the dough into a ring; pinch the edges closed. Use parchment paper to lift the cake and slide a baking sheet beneath the parchment.

Whisk 1 egg with water in a small bowl and using a pastry brush, brush cake with egg wash.

Bake king cake in the preheated oven until golden brown, about 40 minutes. Let cake cool.

Stir 1 cup confectioners' sugar, corn syrup, and milk in a bowl; mix in almond extract. Drizzle king cake with icing and frost with colored sugars.