Combine the yeast and granulated sugar in the bowl of a stand mixer fitted with a dough hook. Add the melted butter and warm milk. Beat at low speed for 1 minute. With the mixer running, add the egg yolks, then beat for 1 minute at medium-low speed. Add the flour, salt, nutmeg, and lemon zest and beat until everything is incorporated. Increase the speed to high and beat until the dough pulls away from the sides of the bowl, forms a ball, and starts to climb up the dough hook. (If the dough is uncooperative in coming together, add a bit of warm water (110 degrees), a tablespoon at a time, until it does.)
Remove the dough from the bowl. Using your hands, form the dough into a smooth ball. Lightly oil a bowl with the vegetable oil. Place the dough in the bowl and turn it to oil all sides. Cover with plastic wrap and set aside in a warm, draft-free place until doubled in size, about 2 hours.
Meanwhile, make the filling. In a large mixing bowl, combine the cream cheese and 1 cup of the confectioner's sugar. Blend by hand or with an electric mixer on low speed. Set aside.
Line a baking sheet with parchment paper.
Turn the dough out onto a lightly floured work surface. Using your fingers, pat it out into a rectangle about 30 inches long and 6 inches wide.
Spread the filling lengthwise over the bottom half of the dough, then flip the top half of the dough over the filling. Seal the edges, pinching the dough together. Shape the dough into a cylinder and place it on the prepared baking sheet seam side down. Shape the dough into a ring and pinch the ends together so there isn't a seam. Insert the king cake baby or pecan half into the ring from the bottom so that it is completely hidden by the dough.
Cover the ring with plastic wrap or a clean kitchen towel and place in a warm, draft-free place. Let the dough rise until doubled in size, about 45 minutes.
Meanwhile, preheat the oven to 350°F.
Brush the top of the risen cake with 2 tablespoons of the milk. Bake until golden brown, 25 to 30 minutes. Remove from the oven and let cool completely on a wire rack.
Make the icing. Combine the remaining 3 tablespoons milk, the lemon juice, and the remaining 3 cups confectioner's sugar in medium-size mixing bowl. Stir to blend well. With a rubber spatula, spread the icing evenly over the top of the cake. Sprinkle with the sugar crystals, alternating colors around the cake.
The cake is traditionally cut into 2-inch-thick slices with all the guests in attendance.