In a large mixer, combine all the ingredients and mix until combined using the paddle attachment. Switch to dough hook attachment; knead dough until smooth and elastic, about 5-7 minutes, adding additional flour (1 tablespoon at a time) as needed to keep the dough from sticking to the sides of the bowl. Place the dough in a greased bowl, over and let rise until doubled in size, approximately 1 1/2 hours.
Remove the dough from the bowl and gently deflate. Divide the dough in half. Roll each half into a long rectangle, approximately 24x6-inches.
Combine the filling ingredients and sprinkle generously over both rectangles.
From the long edge roll the dough tightly and pinch the edges together to create a tight seam. If making the smaller king cakes, cut the strip into 6 inch segments. Take each segment and form into a ring gently pinching the ends together to seal. Place the ring(s) onto a parchment-lined sheet pan(s). Cover and allow to rise for 30-45 minutes.
Preheat oven to 375°F. Place the sheet pan(s) into the oven and bake for 30 minutes or until golden brown. Remove the cakes from the oven and allow to cool on a wire rack.
Combine the icing ingredients until smooth, thinning with additional milk if necessary and drizzle over cooled cake. Sprinkle with colored sugar or sprinkles and serve.