Heat oil in deep-fryer to 375 degrees F (190 degrees C).
Whisk together the egg, milk, and 2 teaspoons of vegetable oil in a bowl until smooth. Stir in the pancake mix until no dry lumps remain. Dip the cookies into the batter one at a time, and carefully place into the hot frying oil. Fry only 4 or 5 at a time to avoid overcrowding the deep fryer. Cook until the cookies are golden-brown, about 2 minutes. Drain on a paper towel-lined plate before serving.