PREPARE THE FILLING
In a small bowl, mix together brown sugar and cinnamon. Set aside.
MAKE THE DOUGH
To a large mixing bowl, add the warm milk, and instant yeast. Whisk together until dissolved.
Then add the sugar, salt, egg, butter, and flour.
Combine together using a hook attachment and stand mixer or a wooden spoon/rubber spatula. Knead the dough for about 3 to 4 minutes and then transfer to an oiled bowl.
Cover, and place in a warm place to rise about 1 hour to 1 hour and a half.
MAKE THE ICING
To a wide bowl, add the powdered sugar, whole milk, and vanilla extract.
Whisk together very well until smooth and creamy.
SHAPE AND FRY
Once the dough has risen, flour a working surface and roll out the dough with a rolling pin into a large rectangle until it is about a little less than 1/2 an inch thick and 14 by 20 inches in width and length,
Add the softened butter on top of the dough and use a rubber spatula to spread the butter into an even layer across the dough.
Sprinkle the brown sugar and cinnamon mixture over the butter and press it down firmly.
Roll it up tightly into a log and pinch the ends together very tightly.
Cut the dough into about 3/4 inch slices to end up with about 16 to 18 pieces. Transfer each roll to an individual parchment paper cut into a small square and place on a baking tray.
Optional, for flatter donuts, smash each roll down a bit. For thicker donuts, leave as is.
Cover and allow it to proof again for about 45 minutes to an hour.
Heat oil suitable for frying in a pot or deep pan like vegetable, canola, or peanut oil. The perfect temperature is around 325°F to 350°F.
Transfer 3 to 4 donuts into the hot oil, making sure not to crowd them. Fry on each side until beautifully golden, flipping over halfway.
Transfer to a wire rack to drain the excess oil.
If the icing has hardened, heat in the microwave for 15 to 30 seconds.
Dunk each donut in the icing, let any excess drip, and place it back on the wire rack to settle.
After 5 minutes and the icing has hardened to the touch, enjoy warm!