Step 1
Mix 4 cups of flour white sugar yeast and salt together in the work bowl of a stand mixer fitted with a dough hook.
Step 2
Whisk 3 eggs in a separate bowl. Melt butter in a saucepan over low heat and whisk buttermilk into melted butter; heat until an instant-read thermometer reads 120 degrees F (49 degrees C). Whisk buttermilk mixture into beaten eggs and cool the mixture to 110 degrees F (43 degrees C).
Step 3
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 1/4 cup more flour.
Step 4
Turn dough out onto a work surface and knead for 1 minute; form into a ball and place into a buttered bowl turning dough around to lightly coat with butter. Cover bowl with plastic wrap and let rise in a warm place until doubled about 2 hours.
Step 5
Preheat oven to 350 degrees F (175 degrees C). Butter the outer edge of an 8-inch cake pan.
Step 6
Stir cream cheese 1 cup confectioners' sugar 2 tablespoons flour lemon juice vanilla extract and nutmeg in a bowl until filling is smooth.
Step 7
Punch down dough and turn out onto a lightly floured piece of parchment paper. Roll dough into a 10x28-inch rectangle. Spread cream cheese filling over the rectangle leaving a 1-inch border along each edge. Lift an edge of the parchment paper sheet to roll dough into a log shape starting at a 28-inch edge. Pinch seams closed to seal in filling keeping roll on the parchment paper.
Step 8
Wrap the filled dough around the outside of the buttered 8-inch cake pan to form the dough into a ring; pinch the edges closed. Use parchment paper to lift the cake and cake pan and slide a baking sheet beneath the parchment. Gently free cake pan leaving the ring-shaped cake. Whisk 1 egg with water in a small bowl; brush cake with egg wash.
Step 9
Bake king cake in the preheated oven until golden brown about 40 minutes. Let cake cool.
Step 9
Stir 1 cup confectioners' sugar corn syrup and milk in a bowl; mix in 1 teaspoon lemon juice or amount needed to thin icing to a thick but slightly runny texture. Drizzle king cake with icing.