Stir the yeast into the milk. Let it sit on the counter until it begins to foam.
Pour the yeasty milk into the bowl of a large mixer. Using the paddle attachment, slowly blend about half of the bread flour into the milk. Mix for three minutes. Stop and scrape down the sides of the bowl.
Add the remaining flour (both kinds), egg, salt and half the melted butter. With the hook attachment, run the mixer on medium-slow speed until the dough rolls up and pulls away from the sides of the bowl, leaving almost no dough sticking to the bowl.
Move the dough to a glass bowl. Add the remaining melted butter and coat the dough with the butter. Cover the glass bowl with a damp, warm towel or plastic wrap. Keep in a warm space for an hour.
When the dough is approximately twice its original size, , place it on a floured board and roll it out into a sheet of dough about a quarter-inch thick. Slice the dough sheet into two sheets lengthwise, about the same size.
Lay down about half of the ham, salami and cheese down half of one of the dough sheets. Leave a narrow, uncovered strip along the edge of the meat and cheese. Fold the uncovered dough over and press it down along the edge, to seal the meat and cheese inside the dough. Repeat this for the second half of the dough, meat and cheese.
On a large cookie sheet lined with parchment paper, connect the ends of the two sealed pieces of dough to form a king cake. Bake the "cake" in a 350-degree oven, and bake at 350 until golden brown.
Remove to a serving platter and ladle the olive salad over the top.
