Working with one at a time, gently fold wonton wrapper diagonally so that corners meet. Repeat with opposite corners to create a small triangle. Using kitchen shears, cut desired shapes into corners and sides of triangle. Transfer to a parchment-lined baking sheet. Keep finished snowflakes in a single layer, covered with plastic wrap or a damp paper towel, while working.

Heat 1 inch oil in a large cast-iron skillet until shimmering. Test oil with a scrap of wonton wrapper; it will sizzle immediately when oil is hot enough to fry. Working in batches so as not to crowd skillet, fry snowflakes until golden, flipping once, 20 to 30 seconds. Transfer to paper towel-lined baking sheets, spreading in single layer. Using a small sieve, dust with confectioners' sugar; serve immediately.