Fit a large Dutch oven with a deep-frying thermometer. Pour in the oil and bring to 325 degrees F over medium-low heat. Line a plate with paper towels.
Position an egg roll wrapper with a corner facing towards you. Place 3 heaping tablespoons mashed potatoes in the center of the wrapper. Cover with a heaping teaspoon Cheddar and use your fingers to mold the filling into a rectangular shape. Fold the bottom corner up over the filling, tucking it to cover the filling. Fold in the left and right corners to meet in the middle. Dampen the corner farthest from you with water, then continue to roll up. Use a little more water to seal the seam. Continue with the remaining wrappers and filling.
Gently drop the egg rolls in batches in the hot oil, seam-side down. Cook, turning the rolls to brown evenly, until deeply golden brown, 3 to 5 minutes. Drain on the paper towels and serve hot.