For Dough:

In a large bowl, using a stand mixer electric mixer, or bread machine, combine butter, milk, eggs and egg yolk, all-purpose flour, sugar, dried milk, salt, yeast, nutmeg and lemon oil.  Mix dough until it forms a silky, very smooth dough.  It will be sticky and soft.

Lightly cover dough and allow to rise in a warm location for 1 hour.  It will puff, but won’t completely double in size.

For Almond Filling:

In a medium sized bowl, beat almond paste, unsalted butter, and almond extract together until well incorporated. Add granulated sugar and white bread crumbs.  Beat again until evenly distributed.  The mixture will be moist and stiff.

Assembling the King Cake:

On a lightly greased surface, roll the dough into a 6″ by 24″ rectangular strip. Pat and spread the almond filling evenly over the dough’s surface, leaving a 1″ strip bare along the longest edge closest to you. Brush this bare edge with a light coat of water.  Roll the dough into a log, starting at the edge farthest away from you. Pinch the long seam closed.  Place the dough log onto a greased baking sheet or a parchment lined sheet.  Use a lightly greased, oven-safe round to form the dough around.  Pinch the ends together.  Cover lightly and allow to rise in a warm location for about an hour. Dough will be puffy.  Preheat oven to 350 degrees F as the dough is rising.

Beat the reserved egg white together with 1 tsp. of water.  Brush over the risen dough.

Bake the cake for 20 minutes.  Tent with aluminum foil and continue baking for 30 more minutes. The dough will develop a deep golden brown color.  Let cake cool on a wire rack for 15 minutes, then moving the cake to the wire rack to cool completely.

For Icing: 

Mix the powdered sugar, milk, salt and vanilla together.  Start with 2 tbsp. of milk and add remaining to sugar until the icing is thick and pourable.  Pour over cooled cake and decorate with colored sugars.