Bring a large pot of salted water to a boil. Add chard, pushing leaves down into the water with a spoon. Blanch for 2 minutes, then strain chard and transfer to an ice bath. When chard is cool, remove from ice bath, and squeeze to remove excess water. Roughly chop chard again, and place
 in a paper towel–lined bowl. Set aside.
Meanwhile, bring 2½ cups of water and ¼ teaspoon of salt to a boil in a medium saucepan. Add barley, reduce heat to medium, and simmer for 25 to 35 minutes. Drain, and spread on a sheet pan to cool. Press paper towels over barley to absorb any extra moisture. Set aside.
Heat 1 tablespoon oil over medium-high heat in a large saucepan. When oil is warm, add mushrooms. Cook, stirring occasionally, for 5 minutes. Reduce heat to medium, and add garlic and 1 tablespoon oil. Cook, stirring often, for 3 to 5 minutes, until garlic is soft and fragrant. Add chard and cook, stirring, for another 2 to 3 minutes. Turn off heat, taste, and adjust seasoning if necessary.
Transfer barley to a large bowl, and toss with remaining oil, lemon zest, and juice. Add mushroom-chard mixture, and stir to combine. Taste and adjust seasoning as needed. Toss in shaved cheese. Serve warm or at room temperature.