To make the dough: Mix flour, sugar, yeast, and salt together in the bowl of a stand mixer fitted with a dough hook. Whisk eggs in a separate bowl.

Melt butter in a saucepan over low heat. Whisk in buttermilk and heat until an instant-read thermometer reads 120 degrees F (49 degrees C). Slowly pour buttermilk mixture into beaten eggs, whisking constantly; let cool to 110 degrees F (43 degrees C).

Beat egg mixture into flour mixture on medium-low speed until a firm, elastic dough forms, about 10 minutes. If dough doesn’t separate from the sides of the mixing bowl during kneading, beat in up to 1/4 cup more flour.

Turn dough out onto a work surface and knead for 1 minute. Shape dough into a ball and place into a buttered bowl, turning dough to lightly coat all over with butter. Cover the bowl with plastic wrap and let rise in a warm place until doubled in volume, about 2 hours.

Preheat the oven to 350 degrees F (175 degrees C). Butter the outside edge of an 8-inch cake pan.

Make the filling: Stir cream cheese, confectioners’ sugar, flour, lemon juice, vanilla, and nutmeg in a bowl until smooth.

Punch down dough and turn out onto a lightly floured piece of parchment paper. Roll dough into a 10×28-inch rectangle. Spread filling over the rectangle, leaving a 1-inch border along each edge. Starting at one long edge of the rectangle, lift the parchment paper sheet and roll dough into a log. Pinch the seams closed to seal in the filling, keeping the roll on the parchment paper.

Wrap the filled dough around the outer edges of the prepared cake pan and pinch the dough ends to form a ring. Use parchment paper to lift the cake and cake pan and slide a baking sheet beneath the parchment. Gently free the cake pan, leaving the ring-shaped cake.

Whisk egg with water in a small bowl to make an egg wash; brush over cake.

Bake in the preheated oven until cake is golden brown, about 40 minutes. Remove from the oven and let cool, about 30 minutes.

Meanwhile, make the icing: Stir confectioners’ sugar, corn syrup, and milk together in a bowl. Mix in 1 teaspoon lemon juice, or as much as needed to thin icing to a thick but slightly runny texture.

Drizzle icing over cooled king cake.