In a large bowl, or the bowl of a stand mixer, gently combine the warm milk with the yeast. Let sit for 8 minutes to bloom.
Once the yeast has bloomed, add the light brown sugar, 2 Tablespoons canola oil, melted butter, salt, and vanilla. Whisk until combined.
Add the egg, whisk until combined.
Sift in the flour, half at a time. Mix by hand or with a dough hook until flour has been incorporated. Dough should be soft, but not overly sticky, add additional flour if needed 1 Tablespoon at a time.
Knead by hand for 5 minutes, or in a stand mixer for 2 to 3 minutes.
Cover and let rise in a warm location for 45 minutes or until doubled in size.
After the dough has risen, fill a medium pot or dutch oven with enough canola oil to fry in (at least 3” deep). Heat the oil over medium heat until it reaches between 355 degrees F and 365 degrees F.
While the oil is heating up, turn the dough out onto a clean and lightly floured surface. Slice the dough into 12 pieces.
Take each piece of dough and form into a ball by pulling the outer edges to the bottom center of the dough ball. Let the balls of dough rest, covered, for 15 minutes.
Fill a shallow dish with 2 cups of sugar and 2 Tablespoons of cinnamon, mix. Set near your frying station along with a wire rack over a baking sheet or paper towels to set the finished fried dough on.
Once the dough has rested and oil has come to temperature, use a rolling pin to flatten a few of the dough balls into oblong shapes, similar to a beaver’s tail. These should be very thin ¼” or less.
One at a time, gently lay the flattened dough into the hot oil. Fry for 30 to 60 seconds on each side, using a pair of tongs or chopstick to flip the dough.
When cooked, pull the beaver tail out and let the excess oil drip for just a second before immediately dunking into the cinnamon sugar mixture. Cover completely, turning over as needed. Set on the wire rack to cool. Give this first one a taste and adjust cooking time or temperature as needed.
Continuously check the temperature of the oil to make sure it is not getting too hot or too cold.
Continue steps 11 to 14 until all of the dough has been fried and sugared.