Take out eggs, butter, and cream cheese to soften.
Heat milk to 100-110°F, pour into the stand mixer bowl, sprinkle yeast over warmed milk, and let sit for 10 minutes.
Grease a large bowl for dough proofing.
Add in eggs, butter, white sugar, and vanilla extract to stand mixer and mix until combined.
Sprinkle in flour, salt, ground cinnamon, and lemon zest. Mix until dough forms a ball and is smooth.
Knead dough on a floured surface until elastic, about 3-5 minutes.
Form into a ball, transfer to the greased bowl, cover with a damp cloth, and let rise in a warm place for 1 hour.
Mix together brown sugar, cinnamon, and nutmeg in a small bowl and set aside.
Combine cream cheese, sugar, vanilla extract, and lemon juice. Beat until smooth.
After proofing, turn dough onto a floured surface and cut in half.
Roll each piece into a rectangle, about 10 x 18 inches and ¼ inch thick.
Spread half of the softened butter onto each rectangle, and sprinkle half of the cinnamon sugar mixture over each rectangle.
Pipe half of the cream cheese filling along one long end of each rectangle.
Roll the dough rectangles into rolls, sealing the seam well.
Stretch each roll to about 24 inches long.
Lay the two rolls side by side on a greased baking sheet and gently twist together, forming an oval.
Cover again and let rise for 30 minutes.
Preheat oven to 350°F. Brush king cake with egg wash, and bake for 30 minutes.
Mix milk, lemon juice, and vanilla together, sift in powdered sugar, and pour icing over the cooled king cake.