Mix 4 cups of flour, white sugar, yeast, and salt together in the work bowl of a stand mixer fitted with a dough hook.
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).
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.
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.
Preheat oven to 350 degrees F (175 degrees C). Butter the outer edge of an 8-inch cake pan.
Stir cream cheese, 1 cup confectioners' sugar, 2 tablespoons flour, lemon juice, vanilla extract, and nutmeg in a bowl until filling is smooth.
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.
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.
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 1 teaspoon lemon juice, or amount needed to thin icing to a thick but slightly runny texture. Drizzle king cake with icing.
