Heat 1 tablespoon of the canola oil in a large skillet over medium-high heat. Add the dumplings and cook until browned, 2 to 3 minutes per side. Add 3 tablespoons water and cook, covered, until cooked through, 8 to 10 minutes.
Meanwhile, heat the remaining canola oil in another large skillet over medium heat. Add the shallots and cook, stirring occasionally, until soft, 5 to 7 minutes. Stir in the edamame, ginger, and 2 tablespoons water. Reduce heat to low and cook, covered, until the edamame are heated through, about 5 minutes. Remove from heat and add the watercress and 1/2 teaspoon salt and toss until just wilted.
Divide among individual bowls and top with the dumplings.
Whisk the sesame oil, rice vinegar, and soy sauce in a small bowl and drizzle over each salad.