Heat vegetable oil in a small skillet over medium-low heat; add minced garlic. Slowly cook garlic until golden brown, about 5 minutes. Remove skillet from heat.
Crush peppercorns using a mortar and pestle until powdery. Add peeled garlic and a pinch of salt; crush into a paste. Mix cilantro stems into the paste.
Mix paste, pork, shrimp, water chestnuts, dark soy sauce, 1 tablespoon white sugar, fish sauce, tapioca starch, light soy sauce, and 1/2 teaspoon salt together in a large bowl. Spoon 1 to 2 teaspoons pork-shrimp mixture into the center of each wonton wrapper. Fold the wrapper over the filling, sealing the edges together and creating a purse-like pouch.
Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add wontons, cover, and steam until the filling is cooked through, about 10 minutes.
Whisk rice vinegar, light soy sauce, 1 tablespoon white sugar, scallions, and bird's eye chile together in a bowl until dipping sauce is well mixed.
Serve Khanom Jeeb with a drizzle of garlic oil and dipping sauce on the side.