Pre-heat a frying pan with a tight fitting lid over medium heat. Non-stick won't brown the gyoza as well as a regular pan but they do have a tendency to stick a bit so decide what your priorities are.
Add 1 tsp vegetable oil and 1 tsp sesame oil.
Making sure you don't let the gyoza touch, carefully place as many as will reasonably fit in the pan and fry, flat side down (pleated side up if you bothered) for about 2 minutes. Check to see how they are browning.
When they are just a bit less brown than you want, add a 1/4 cup of water. Be careful. You are adding water to hot oil. That's generally not a good idea. It will splatter.
Cover, turn the heat to medium-low and cook for about 2-3 minutes.
Remove the lid, turn the heat back to medium and cook until the water has evaporated.
If you are into food safety, check the internal temperature with an instant read thermometer. You want the inside to read at least 160F. I've never seen it not get there but it's always a good idea to check.
Sprinkle with togarashi shichimi if you have it. Serve with dipping sauce.