Whisk the yeast with the warm milk in the bowl of a stand mixer until dissolved. Add 6 Tbs. bread flour and the honey. Using the paddle attachment, mix on low speed until smooth with a few small lumps, scraping the bottom and sides of the bowl with a rubber spatula. Cover the bowl with plastic wrap and let it rise until doubled in volume, about 20 minutes.
When the dough is doubled, add 3/4 cup bread flour, and all the cake flour, eggs, egg yolk, sugar, cinnamon, vanilla and almond extracts, and salt. Mix on low speed until combined, then switch to a dough hook. Increase the speed to medium, and beat until smooth, about 2 minutes.
Coat the inside of a large bowl with 1/2 Tbs. of butter. Transfer the dough to the bowl, turning it over in the bowl to coat with butter. Cover the bowl with a piece of plastic wrap and put it in a warm, draft-free spot until the dough has doubled in size--about 1 hour.
Line a rimmed baking sheet pan with parchment paper. Coat the parchment paper with the remaining butter. Generously flour your work surface with bread flour. Turn the dough out onto the work surface and sprinkle the top with flour. Using your hands, press and flatten it into a rectangle.
With a rolling pin, roll the dough into a 1/4-inch-thick strip about 24 inches long by about 6 inches wide. Starting with one of the long sides, roll the dough on top of itself, making a long, thin baguette-shaped length. Pinch the edge to the body of the dough to seal. Then turn the dough so it lies horizontally on your work surface, and gently roll it to even out any bulges and create a more or less consistent 1 1/2-inch-wide rope. Bring the two ends of the dough together and pinch them together.
Carefully transfer the dough oval or circle to the prepared sheet pan. Cover with a piece of plastic wrap. Set in a warm, dry spot to rise until doubled, about 1 hour.
Heat the oven to 375°F. Whisk the egg and milk together in a small bowl. Brush the egg wash over the top and sides of the dough, and bake the king cake until golden and cooked through, 25 to 30 minutes. Set on a rack to cool completely.
Make the icing while the cake cools. Whisk the confectioners’ sugar, corn syrup, milk, and vanilla together in a bowl until smooth and completely incorporated. Cover the bowl with a damp kitchen towel until you are ready to glaze the cake.
