In a large mixing bowl, combine the shredded cabbage, crumbled tofu, chopped shiitake mushrooms, minced garlic, grated ginger, and chopped basil. Drizzle in the soy sauce, and mix everything together until well combined. Set aside.
In a small bowl, mix the cornstarch with cool water until smooth. This will be your sealing paste that helps keep the dumplings closed during cooking.
Take a dumpling wrapper and place about a tablespoon of the filling in the center. Using your finger, dab a bit of the cornstarch mixture around the edges of the wrapper. Fold the wrapper in half to create a half-moon shape, pressing the edges together to seal. You can crimp the edges for a decorative touch if you like! Repeat this process with the remaining wrappers and filling.
Heat the cooking oil in a large frying pan over medium heat. Once hot, arrange your potstickers in a single layer in the pan. Cook for about 2-3 minutes until the bottoms are golden brown.
Carefully add about 1/4 cup of water to the pan (be cautious of splattering) and immediately cover with a lid. Allow the potstickers to steam for about 5-7 minutes, or until the water has evaporated and the potstickers are cooked through.
Remove the potstickers from the pan and serve immediately with your favorite dipping sauce. Enjoy the crispy bottoms and tender filling!