Use a circular cookie cutter to cut circles out of each wonton wrapper. Use the largest cookie cutter that will fit inside the square wrapper—you want circles that are as large as possible. Set aside the cut circles and cover with a damp paper towel.
In a medium sized bowl, combine the turkey, kimchi, green onions, and beaten egg. Toss to combine.
Prepare a small bowl of water. Place one circle on a flat surface. Dip your finger into the water and moisten the edge of one half of the circle. Place 1 teaspoon of filling in the middle of the circle. Bring the edges together, creating a half moon. Starting at one end, pinch the edge closed. Fold the wonton wrapper 4–5 times across the rounded edge of the half moon. Pinch the final edge to seal.
Place on a flat surface and squish the potsticker so that it sits flat (folded edge up). Repeat with remaining wonton wrappers and filling.
Heat oil in a medium sauté pan over medium heat. Once hot, add a layer of potstickers, leaving about 1 inch between potstickers. Sauté for about 2 minutes, or until the bottom of the potstickers is light brown.
Add ¼ cup water (careful—it will sizzle when you add it!). Cover and reduce the heat to low. Cook for 3–4 minutes, or until the wonton wrappers are slightly translucent. Uncover the pan and turn heat back to medium. Cook until the water is evaporated, about 1–2 minutes.
Remove the potstickers from the pan and place on a paper towel lined plate. Repeat with remaining potstickers.
Potstickers can also be frozen and cooked as listed above, increasing the steaming time to 5–6 minutes instead of 3–4 minutes.
While the potstickers are cooking, place the dipping sauce ingredients in a jar with a tight fitting lid. Shake to combine. Serve as a dipping sauce for freshly cooked potstickers.