1. In a large nonstick skillet, heat 1 teaspoon vegetable oil over medium-high heat. Add chard, mushrooms, scallions, and garlic and cook until chard stems are crisp-tender and liquid has evaporated, 3 to 4 minutes. Squeeze excess liquid from chard. Transfer to a bowl; stir in vinegar, sesame oil, and soy sauce; season with salt and pepper. Wipe skillet dry and set aside.

2. Place 1 rounded teaspoon filling toward the edge of a dumpling wrapper. Moisten one edge with water using your finger. Bring two edges together, forming a half moon and enclosing filling. Pinch a couple of small pleats along top layer of wrapper. Seal by pressing pleated and unpleated edges together. Place on a parchment-lined baking sheet. Repeat. Keep dumplings and unused wrappers covered with damp paper towels while working.

3. Heat remaining tablespoon vegetable oil in reserved skillet over medium-high heat. Arrange dumplings in pan, tightly, flat-side down, and cook until deep golden brown, 2 to 3 minutes. Remove from heat. Add 1/2 cup of water. Return to heat. Cover and cook until heated through, all the water has evaporated, and bottoms are crisp, 4 to 6 minutes more (remove the lid for the last 30 seconds of cooking). Slide dumplings onto a serving platter, and serve with dipping sauce.