Place butternut squash cut-side down in a shallow dish; cover with plastic wrap.
Microwave squash until flesh is easily pierced with a fork, about 12 minutes. Cool until easily handled, about 5 to 10 minutes.
Scoop squash flesh into a large bowl, discarding skin. Mix brown sugar and 2 tablespoons soy sauce into the squash. Stir in scallions and ginger.
Spoon 1 tablespoon squash mixture into the center of a wonton wrapper. Moisten edges with water and fold into a triangle. Repeat with remaining wonton wrappers. Arrange filled potstickers on a baking sheet.
Freeze potstickers until almost solid, about 15 minutes.
Heat 2 tablespoons oil in a large skillet over medium heat. Add a layer of potstickers; cook, shaking skillet occasionally, until bottoms are golden, about 2 minutes. Flip and add 1/2 cup water. Cover and let steam until cooked through, about 3 minutes. Transfer to a serving dish. Repeat with remaining oil and potstickers.
Mix 1/2 cup soy sauce, water, and sesame seeds together in a small bowl to make dipping sauce. Serve alongside potstickers.