Whisk the milk, eggs, and butter in the bowl of a stand mixer.
Add the flour, sugar, yeast and salt. Stir with a spoon until just combined.
Using the dough hook, mix the dough with an electric mixer for 10 minutes.
Remove the dough from the stand mixer bowl and transfer to a greased bowl. Cover with plastic and allow to rise for 1 ½ to 2 hours.
Add the Filling:
Preheat the oven to 350 degrees Fahrenheit.
In a small bowl, combine the brown sugar and cinnamon. Mix to combine.
Place the dough on a lightly floured surface and roll into a rectangle until dough is around ¼ inch thick.
Spray the dough lightly with water and sprinkle the brown sugar mixture all over the dough, pressing it into the dough with your fingers to ensure it sticks.
Using a knife or pizza cutter, cut the dough in half lengthwise. Then cut the dough horizontally into 2 inch strips.
Roll each strip lengthwise, then form a circle with the rolled dough and seal the ends together. Place each roll into the basin of a greased mini Bundt cake pan with the seam of the roll facing up. Continue rolling each strip until each space is filled.
Bake 20 to 25 minutes or until the mini king cakes are a deep golden brown.
Let the cakes cool in the pan for 10 minutes, then transfer them to a cooling rack to cool completely.
Make the Glaze:
While the cakes are cooling, place the powdered sugar and milk in a bowl.
Whisk the ingredients together until smooth.
Spoon the glaze over the top of each cake. Immediately sprinkle purple, green, and yellow sanding sugar on top of the cake.