Cheesy Barley & Chard

2 tablespoons extra-virgin olive oil
2 cups chopped shiitake-mushroom caps (5 ounces)
1 bunch Red Dog chard, stems thinly sliced, leaves chopped
Kosher salt and freshly ground pepper
2 teaspoons minced garlic (from 2 cloves)
1 cup pearled barley
1 can (28 ounces) diced tomatoes
2 ounces Parmigiano-Reggiano, grated (1/2 cup)
1 tablespoon fresh lemon juice or red-wine vinegar
3 ounces Gruyère or white cheddar, grated (1 cup)
Fresh basil leaves, for serving

Heat oil in a large straight-sided skillet or braiser over medium-high. Add mushrooms and chard stems; season with salt. Cook, stirring occasionally, until mushrooms are tender and golden, 5 to 6 minutes. Stir in garlic; cook 30 seconds. Add barley and cook, stirring, until toasted, about 1 minute.

Stir in tomatoes and 1 cup water; season with salt and pepper. Reduce heat to medium-low, cover, and simmer, stirring occasionally, until barley is tender and creamy, 25 to 30 minutes.

Stir in chard leaves, Parmigiano, and lemon juice; cover and cook, stirring once, until chard is wilted, 2 to 3 minutes. Season, top with Gruyère, cover, and heat until melted, 2 minutes. Serve with basil.