Mix chicken, 2 tablespoons scallions, and 1 tablespoon soy sauce together in a bowl. Season with salt and pepper.
Place 1/2 teaspoonful of chicken mixture in the center of each wonton wrapper. Dip your finger in water and run it along the edge of each wrapper. Bring corners together in the center and press to seal. Pinch edges together.
Heat sesame oil in a large skillet over medium-high heat. Cook pot stickers, in batches if necessary, until browned on the bottom, 2 to 3 minutes. Reduce heat to medium; pour in chicken broth and cover skillet. Steam pot stickers until opaque, 5 to 6 minutes. Drain any excess broth.
Mix remaining 2 tablespoons scallions, 1/4 cup soy sauce, teriyaki sauce, and chile oil together to make dipping sauce. Serve sauce alongside pot stickers.