In shallow bowl, whisk 2 tablespoons tamari, Worcestershire sauce, vinegar, sesame oil, shallots and mustard. Add mushroom caps and marinate 15 minutes, turning occasionally.
Meanwhile, remove stems from chard and coarsely chop.
In large skillet, heat 1 teaspoon oil over medium heat. Add garlic and cook, stirring, 1 minute. Add chard and remaining tamari. Cover and cook until wilted, 4 minutes. Uncover and set aside.
Wipe out pan. Heat remaining 1 1/2 teaspoons oil over medium-high heat. Remove mushrooms from marinade; reserve marinade. Add mushrooms and cook until tender, 4 minutes per side.
To serve, reheat chard and divide among plates. Cut mushrooms into 1/2- inch-thick slices. Arrange sliced mushrooms over chard and drizzle with remaining marinade if desired.