Prep Filling: Add mashed potatoes and chopped turkey to a medium-sized bowl and mix well. Crack the egg into a small bowl and whisk. Add cranberry sauce to another small bowl and prepare a clean work surface or cutting board to assemble the wontons on. Prep a kitchen towel or some paper towels to clean hands with while making wontons.
Fill Wontons: Take 1 tablespoon of the mashed potato turkey filling and add to the center of a wonton wrapper. Top with 1/2 teaspoon of cranberry sauce. Dip a finger or a small pastry brush into the egg wash and egg wash one side of the wonton wrapper. Repeat for all 4 sides.
Fold Wontons: Bring two corners of the wonton wrapper up to meet, but don't seal them yet. Gently press the sides of the wonton to the sides of the filling. Bring the other two corners up to meet in the center. Gently press all 4 corners together, sealing at the center. Squeeze around the filling gently to release any air and then gently press together to seal the edges so that the wonton is fully sealed.
Fry: Heat vegetable or canola oil in a medium saucepan on high until it reaches 350℉ (use an instant-read thermometer or wooden spoon test, see recipe notes). Add 3-4 wontons to the oil and fry for 30-45 seconds, turning if needed so all sides are golden. Remove from oil and let cool on a paper towel lined plate. Repeat for all wontons.
Serve: Serve with warmed leftover gravy, or make a quick batch (see recipe notes).