Bring a large pot of salted water to a boil. Carefully remove the large outer leaves from the cabbage head. Submerge the leaves in the boiling water for 2 to 3 minutes until they become pliable and soft. Immediately transfer the leaves to a bowl of ice water to stop the cooking process. Drain them well and pat them dry with a kitchen towel. If the center rib of the leaf is too thick, use a paring knife to shave it down slightly so it rolls easier.

In a large mixing bowl, combine the ground meat or tofu, minced mushrooms, garlic, ginger, green onions, soy sauce, sesame oil, and pepper. Use your hands or a sturdy spoon to mix the ingredients thoroughly. Ensure the aromatics are evenly distributed so every roll has a consistent flavor. Let the mixture sit for 10 minutes to allow the flavors to meld.

Lay one prepared cabbage leaf flat on a clean surface. Place about 2 tablespoons of the filling near the stem end of the leaf. Fold the sides of the leaf over the filling, then roll it tightly from the bottom to the top, similar to a burrito. Repeat this process until all the filling is used. Place the rolls seam-side down to keep them from unraveling.

Place a steamer basket over a pot of simmering water. Arrange the cabbage rolls in the basket in a single layer. Cover and steam for 15 to 20 minutes, or until the internal temperature of the meat reaches 165°F (74°C). Alternatively, you can place them in a shallow pan with half a cup of chicken broth, cover, and simmer on the stovetop for the same amount of time.

Whisk together the sauce ingredients in a small bowl. Transfer the warm cabbage rolls to a serving platter. Drizzle the prepared sauce over the top and garnish with extra green onions or toasted sesame seeds. Serve immediately while hot.