Preheat the broiler. Heat the vegetable oil in a large nonstick skillet over medium-high heat. Add the dumplings flat-side down and cook, undisturbed, until slightly browned on the bottom, about 3 minutes. Add 1/4 cup water to the skillet; cover and cook until the water is evaporated and the dumplings are crisp on the bottom, about 3 minutes. Transfer the dumplings to a foil-lined baking sheet.
Top each dumpling with 1 teaspoon pizza sauce, 1 slice pepperoni and 1 slice mozzarella. Broil until the cheese is melted, about 1 minute. Top with basil. Serve with more sauce for dipping.