Finely chop the broccoli florets into very small pieces so they blend easily into the filling and cook quickly.
In a large mixing bowl, combine the ground beef and chopped broccoli, breaking up any clumps of meat with your hands or a spoon.
Add the soy sauce, oyster sauce, sesame oil, grated ginger, minced garlic, sliced green onions, cornstarch, salt, and black pepper to the bowl.
Mix the filling thoroughly with clean hands or a sturdy spoon until everything is evenly combined and the mixture feels slightly sticky and cohesive.
Place a dumpling wrapper on a clean work surface and keep the remaining wrappers covered with a slightly damp cloth to prevent drying.
Spoon about 1 to 1 1/2 teaspoons of the beef and broccoli filling into the center of the wrapper, keeping the edges clear.
Lightly moisten the outer edge of the wrapper with water using your fingertip, then fold the wrapper in half over the filling to form a half-moon shape.
Press the edges firmly together to seal, gently pushing out any trapped air, and crimp or pleat the edge if desired for a decorative finish.
Repeat the filling and sealing process with the remaining wrappers and filling, arranging the formed dumplings on a plate or tray without overlapping.
Heat 1 tablespoon of vegetable oil in a large nonstick skillet over medium-high heat until the oil is hot and shimmering.
Arrange half of the dumplings in the skillet in a single layer, flat side down, and cook for 2 to 3 minutes until the bottoms turn golden brown.
Carefully pour in 1/2 cup of water, immediately cover the skillet with a lid, and reduce the heat to medium to allow the dumplings to steam.
Steam the dumplings for about 6 to 8 minutes, or until most of the water has evaporated and the wrappers look tender and slightly translucent.
Remove the lid and continue cooking for 1 to 2 minutes to re-crisp the bottoms, then transfer the dumplings to a serving plate.
Repeat the cooking process with the remaining 1 tablespoon of vegetable oil, 1/2 cup of water, and the rest of the dumplings.
In a small bowl, stir together the rice vinegar, soy sauce (for dipping), sesame oil (for dipping), and sliced green onions (for dipping) until well combined.
Serve the hot beef and broccoli dumplings immediately with the dipping sauce on the side for drizzling or dipping.