In a medium pot of boiling salted water, cook barley according to package instructions. Drain and let cool.
Meanwhile, preheat oven to 450 degrees. Place scallions and corn on a rimmed baking sheet. Toss with 1 tablespoon oil and season with salt and pepper. Roast until vegetables are tender, about 25 minutes, stirring halfway through.
In a large bowl, combine barley, roasted vegetables, tomatoes, parsley, chicken, 1 tablespoon oil, and 1 tablespoon lime juice; season with salt and pepper. In a medium bowl, toss spinach with 1 tablespoon oil and 2 teaspoons lime juice; season with salt and pepper. Serve spinach with chicken-barley mixture.
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