Combine the filling: in a large mixing bowl, add the shredded cabbage, crumbled tofu, chopped shiitake mushrooms, minced garlic, grated ginger, chopped basil, and soy sauce. Mix thoroughly until evenly combined.
Make the cornstarch slurry: in a small bowl, whisk 1 teaspoon cornstarch with 1/4 cup cool water until smooth; set aside.
Prepare wrappers: work with one dumpling wrapper at a time. Place a heaping teaspoon of filling in the center of the wrapper.
Seal dumplings: dip a finger in the cornstarch slurry and paint a thin line around the edge of the wrapper, fold into a half-moon, and pinch firmly along the edge to fully seal. Repeat until all wrappers are filled. Lay finished potstickers in a single layer and cover with plastic to prevent drying.
Cook first batch: heat 1 tablespoon oil in a nonstick skillet over medium-high heat. When the oil is hot, arrange potstickers in a single layer, flat side down, and cook until the bottoms are golden brown, about 2 minutes.
Flip and brown: turn each potsticker to brown the other side, about 1–2 minutes more.
Steam: using the pan lid as a shield, add 2 tablespoons water to the pan and cover immediately to steam. Cook 3–4 minutes, until most of the water has evaporated and the filling is cooked through.
Finish and repeat: uncover and remove cooked potstickers. Wipe the pan dry if needed, add more oil, and repeat with remaining dumplings in batches.