Place a large skillet over medium heat.

Add sesame oil.

When hot, add cabbage, onion, garlic and carrot. Cook, stirring, until cabbage is limp.

Add ground pork and egg. Still stirring, cook until pork is evenly browned and egg is no longer runny.

Transfer mixture to a large plate or bowl.

Wipe out skillet with a paper towel. Set aside.

Lay out all wrappers (or as many as you have room for) on a work surface.

Spoon about 1 tablespoon of the cabbage and pork mixture onto the center of each wrapper.

Fold wrappers in half over filling, sealing edges with moistened fingers.

Add vegetable oil to skillet and place back over medium heat.

When oil is hot, cook gyoza about 1 minute per side, until lightly browned.

Place water into skillet and reduce heat. Cover and allow gyoza to steam until the water is gone.

(Cook all gyoza at once, or cook stages if skillet is not big enough. Wipe out with a paper towel, add more vegetable oil and heat until hot between each stage.)