First we will begin making the dough. Add warmed milk to a large bowl, along with 3¾ cups of the flour, 2 tablespoons of oil, salt, sugar, egg, and yeast. Stir together with a wooden spoon until combined. The dough will be very thick but still pretty wet and sticky.
Place 2 tablespoons of flour on the surface of your counter. Place dough on top of it and begin kneading the dough with the flour until all the flour is mixed into the dough. Keep kneading for about 4-5 minutes until the dough is smooth and elastic. If it starts to stick to the counter, add a little more flour. Once you finish kneading, shape into a ball. Lightly grease another large bowl and place the ball of dough into bowl. Cover with a kitchen towel and let it rise for 1 hours, where it will be double the size. I like to place mine in the microwave to keep it away from any cool draft from the A/C.
Once dough has risen, punch it down with your fist and remove from the bowl. Lightly dust your counter with flour and place the dough on top. Roll out dough into a large rectangle, 18 inches x 22 inches. Spread butter over the dough in one even layer. Sprinkle with cinnamon sugar mixture .
Starting on the long edge, tightly roll the dough into a log. Cut the edges off and then cut the log in half and then into quarters so you have 4 parts. Cut each part into 4 even rolls so that you have 16 rolls total. Place onto a large baking sheet, tucking the seam to ensure it stays in one round roll. Cover and let them rise for another 45 minutes or until doubled in size. Preheat cooking oil in a Dutch oven to 350 degrees while the rolls rise.
Once they're done rising and oil is 350-360 degrees, uncover them, place 3 into the oil to fry, making sure the expressed seam is facing up (this will ensure the cinnamon rolls don't unravel while frying). Fry for 1 minute then flip to cook for another minute. Remove from oil and let drain on a rack on top of a baking sheet. Repeat with the rest of the doughnuts.
Once the doughnuts are all fried, it's time to dip them in icing. Place powdered sugar and milk in a medium bowl. Whisk together until smooth. Dip the top of each cinnamon roll doughnut into the icing and place back on the rack to let the excess drip off. Serve warm or within 12 hours.