Sprinkle the beef with salt and pepper. Add the oil to a wok or a large cast-iron or saute pan over high heat. Once heated, add the seasoned beef and sear, stirring quickly, until cooked through, about 2 minutes, then remove from the wok and set aside. In the same pan, add the onions, garlic and ginger over medium heat and cook until caramelized, about 5 minutes. Season with salt and pepper. Add the hot onions to the cooked beef. Taste for seasoning, adjusting if needed. Store in the fridge to cool, at least 20 minutes or up to 1 hour. Fold in the cheese prior to making dumplings.
To form potstickers, line a sheet tray with parchment paper and set aside. Lay out a dumpling wrapper on your work surface. Using your finger or a pastry brush, wipe water around the top edge of the wrapper. Spoon about 1/2 tablespoon filling in the center. (Make sure not to get filling on the edges of the wrapper, which prevents proper sealing.) Fold the wrapper in half to form a half-moon shape. Seal the top center by pressing between fingers and starting at the center, then making three pleats, working toward the bottom right corner. Repeat, working toward the bottom left corner. Press the dumpling gently on a work surface to flatten the bottom. Repeat with the remaining wrappers and filling. Hold the dumplings on the lined sheet tray.
Heat a large saute pan over high heat. Add 2 tablespoons oil and swirl to coat. When the oil shimmers, add the potstickers, flat-side down, in 2 snug rows of 5 and cook, in batches, without disturbing, until the bottoms have browned, 4 to 6 minutes. Add about 1/2 cup water and immediately cover the pan to avoid splattering. Lift the cover and make sure about 1/8 inch water remains; if not, add a bit more. Steam until the potstickers are puffy, yet firm, and the water has evaporated, 8 to 10 minutes. If the water evaporates before the potstickers are done, add more in 2- to 3-tablespoon increments. If the potstickers seem done but water remains, drain the water and return the pan to the stovetop. Continue to cook over high heat to allow the potstickers to re-crisp on the bottom, 2 to 3 minutes. Transfer to a platter and serve hot with Dijon mustard.