Stir together the warm milk and sugar in the bowl of a stand mixer. Stir in the yeast and let sit for 10 minutes, it should bubble and get frothy. Next, add the eggs, sugar, spices, salt, and orange zest and whisk until well combined. Add 3 cups of flour and mix on low with the dough hook attachment until the dough comes together. Add the butter in 1 tablespoon at a time, mixing in between each addition. Knead the dough for 5 minutes on medium-low speed, it should be smooth, stretchy, and still slightly sticky. Place in a greased bowl, cover loosely, and let rise for 1 hour.

Punch down the dough to deflate after it has risen and cut it in half and then cut each half into 8 equal pieces, making 16 in total. Shape each piece of dough into a bun and place 1 ½ inches apart on a large baking sheet lined with parchment paper OR arrange in a greased 9X13-inch baking pan for soft-edged buns. Cover and let rise for another 30 minutes.

Preheat the oven to 400℉. Whisk together the egg and water to make an egg wash. When the buns are done rising, gently cut a cross into the top of each bun with a sharp knife, brush the tops with the egg wash, and bake for 10-14 minutes or until golden. While the buns are cooling down slightly, whisk together the melted butter, powdered sugar, and milk to combine. Fill a sandwich bag with the icing, snip off a small hole in the corner, and pipe into the cross cuts on the mostly cooled buns. These buns are great served while still warm. Enjoy!