Mix brown sugar and cinnamon in a small bowl. Set aside.

In a large mixing bowl, whisk together the warm milk and instant yeast until dissolved.

Add the sugar, salt, egg, butter, and flour.

Mix and knead the dough using a stand mixer with a hook attachment or a wooden spoon and rubber spatula for 3-4 minutes.

Transfer the dough to an oiled bowl, cover, and let rise in a warm place for 1 to 1 1/2 hours.

In a wide bowl, whisk together powdered sugar, whole milk, and vanilla extract until smooth and creamy.

Once the dough has risen, flour a working surface and roll out the dough into a large rectangle about 14 by 20 inches and less than 1/2 inch thick.

Spread softened butter evenly over the dough using a rubber spatula.

Sprinkle the brown sugar and cinnamon mixture over the butter and press it down firmly.

Roll the dough tightly into a log and pinch the ends together.

Cut the log into 3/4 inch slices to get 16 to 18 pieces. Place each roll on a small square of parchment paper and put them on a baking tray.

For flatter donuts, gently press down each roll. For thicker donuts, leave them as is.

Cover and let them proof again for 45 minutes to 1 hour.

Heat vegetable oil in a pot or deep pan to 325°F-350°F.

Fry 3-4 donuts at a time, flipping halfway, until golden brown on both sides.

Transfer the donuts to a wire rack to drain excess oil.

If the icing has hardened, microwave it for 15-30 seconds.

Dunk each donut in the icing, let the excess drip off, and place them back on the wire rack to set.

Once the icing hardens, enjoy your warm cinnamon roll donuts!