In a bowl of a stand mixer fitted with a paddle attachment, mix the flour, sugar, brown sugar, baking powder, salt and cinnamon.
Add the oil, milk, sour cream, eggs, and vanilla. Mix on medium-low speed for 2 minutes, until a smooth and very sticky dough forms.
Scrape down the sides of the bowl, cover with plastic wrap, and let the dough chill in the fridge for 1 hour.
Mix the cinnamon-sugar coating ingredients together in a medium bowl. Set aside for later.
In a large heavy bottom pot, heat 2 inches of oil over medium-high heat until it reaches 365°F when measured with an instant-read thermometer. Once the desired temperature is achieved, reduce the heat to medium-low to maintain the consistent temperature.
Carefully place the carrot nub into the hot oil with a slotted spoon. This will prevent the donuts from burning in the oil and keep the oil nice and clean.
While the oil is heating up, on a lightly floured surface, roll out the dough to ½-inch in thickness, and use a 3 ½-inch donut cookie cutter to cut out donut shapes. Save the donut holes and fry them up, too!
Carefully lower each donut into the pot with a slotted spoon (about 3 donuts can fit at once with some donut holes) and cook for about 2 minutes on each side until lightly golden brown. Cook donut holes for about 1 minute on each side.
Use a slotted spoon to transfer the hot donut to a paper-towel lined plate or pan to cool for 5 minutes.
Roll each warm donut into the cinnamon-sugar mixture to coat.