For the sauce: Whisk together the chili sauce, ketchup, oyster sauce and soy sauce in a small bowl until combined.
For the omelets: Whisk 1/4 cup cold water with the potato starch in a small mixing bowl until combined.
Working in batches, make one omelet at a time by mixing together one-quarter of the beaten eggs, one-quarter of the starch mixture and 4 or 5 oysters in a small bowl. Make sure to whisk the starch slurry frequently to keep it combined. Stir in 1/4 of the basil.
Heat a small wok or 6-inch nonstick skillet over medium heat until it begins to smoke and add about 1 teaspoon of the peanut oil. Place 1 mu shu pancake into the wok and cook until hot and beginning to brown, about 30 seconds, and then flip and toast the other side. Transfer the pancake to a plate. Pour the egg mixture into the bottom of the wok and swirl to coat the base of the pan. Cook, releasing the edges of the omelet with a spatula, until the bottom is set and beginning to brown and the egg on the surface is just set, 30 to 45 seconds. Do not overcook.
Slide the omelet directly onto the toasted warm pancake and drizzle some of the sauce over the top. Repeat with the remaining egg mixture and pancakes to make 3 more omelets. Cut each omelet into quarters and serve face-up, or fold in half and serve with more sauce on the side.