Cook the barley in a medium pot of boiling salted water until tender, about 50 to 60 minutes for hulled or hull-less, 20 to 30 minutes for pearl barley. Drain, then spread out on a baking sheet and let cool.

Meanwhile, cook the sliced shallots in the vegetable oil in a small saucepan over medium-high heat, swirling pan occasionally to keep shallots from burning, until golden brown, 5 to 7 minutes.

Using a slotted spoon, transfer the fried shallots to paper towels to drain; season with salt. Let cool. Set shallot cooking oil aside.

Heat the olive oil in a large skillet over medium-high until just beginning to smoke. Arrange the mushrooms in the skillet in a single layer and cook, undisturbed, until undersides are golden brown, for about 3 minutes. Season the mushrooms with salt and pepper, toss, and continue to cook, tossing often and reducing heat as needed to avoid scorching, until golden brown all over, about 5 minutes longer.

Reduce the heat to medium and add thyme sprigs, garlic, and butter to skillet. Tip skillet toward you so butter pools at edge and use a spoon to baste mushrooms with foaming butter; cook until butter smells nutty. Using a slotted spoon, transfer the sautéed mushrooms to a small bowl, leaving the thyme and garlic behind.

Toss the cooled barley, cilantro, parsley, lemon juice, 1½ oz. Parmesan, and 2 tablespoons of the reserved shallot oil in a large bowl to combine; season with salt and pepper. Add the mushrooms; toss again to combine.

Just before serving, top with the fried shallots and more shaved Parmesan.

Barley can be cooked 1 day ahead. Let cool; store airtight and chill.

Dish can be made 3 hours ahead; store tightly wrapped at room temperature. Do not add the fried shallots and additional Parmesan until right before serving.