Anybody who's ever lived in or near an Italian-American neighborhood and attended a street fair has seen funnel cakes: hot discs of fried dough, crunchy on the outside and soft and warm inside. Powdered sugar is the most popular finishing touch, but I prefer to top them with cinnamon sugar. Whichever you choose, be sure to hit the funnel cakes with it the moment they come out of the oil, so that the sugar melts onto the pastry.
Put the pancake mix in a bowl. In another bowl, whisk the milk, 2 tablespoons of the oil, the egg, yolk, and vanilla. Pour this mixture over the pancake mix and stir to combine well. Put the batter in a measuring cup with a spout or similar vessel, such as a small pitcher.
Line a plate with paper towels. Pour oil into a cast iron skillet to a depth of about 1/2 inch. Heat the oil over medium heat to a temperature of 325 degrees F to 350 degrees F.
Hold a funnel over the oil, a couple of inches from the surface.
Working very carefully to avoid splashing hot oil, pour about one-quarter of the batter into the funnel and, working from the center, drizzle a web of batter over and around to create a disk approximately 6 inches in diameter. It's desirable to have some gaps or holes in the disk, but it should be whole enough to facilitate turning it over.
When the bottom of the funnel cake is lightly browned, about 1 minute, turn the cake over, moving it away from you, and cook until golden on the other side, about 1 more minute.
Transfer the funnel cake to the paper-towel–lined plate and dust immediately with sugar.
Repeat with the remaining batter, letting the oil reheat between cakes, and serve immediately.