Add 3 inches of vegetable oil to a pot set over medium-low heat. Attach the deep-fry thermometer to the pot. Line a plate with paper towels and set it aside.
Arrange 12 of the wonton wrappers on your work surface then place a square piece of chocolate in the center of each wonton.
Brush the egg around the edges of each of the wonton wrappers then place a second wonton wrapper on top and firmly seal together the edges with your fingers.
Once the oil reaches 350°F, add one or two of the wontons to the pot. Cook the wontons, flipping them as needed, until they are golden brown on both sides.
Using a slotted spoon, transfer the wontons from the pot to the paper towel-lined plate.
Repeat the frying process with the remaining wontons, allowing the oil to return to 350°F between each batch.
Dust the wontons with confectioners' sugar and serve immediately.
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