Heat butter in a medium skillet. Brown the chicken thighs for a couple of minutes on the skin-side in butter. Flip over and place the chile, garlic cloves, star anise, and lime juice in the skillet. Cover the pan and cook over medium-low heat for 30 to 35 minutes or until chicken is very tender.
While the chicken cooks, prepare the gyoza wrappers by heating a skillet over medium-high heat. Brush both sides of the wrappers with the sesame oil. Place the wrappers in the hot pan a few at a time until just beginning to form bubbles on the surface and brown. Flip over and do other side. Place on a rack to cool and continue with the remaining wrappers. Set aside.
Combine the sauce ingredients together and set aside. Combine the wasabi cream ingredients together and refrigerate until ready to serve.
Remove the chicken from the cooking liquid. Using 2 forks, shred the chicken. Place the chicken back in the pan with just enough cooking liquid to keep it moist. Place over very low heat to keep warm.
Combine the scallions, cucumber, and snow peas. Toss with a little of the prepared sauce.
To serve, drizzle some of the sauce on each serving plate. Place 1 of the gyoza wrappers on a serving plate. Top with some of the watercress. Layer on some of the shredded chicken. Add another pot sticker wrapper to the stack. Top with some of the vegetables. Add the last wrapper to each stack. Top with wasabi cream and additional fresh scallions, if desired.