1. In a large pot, bring salted water to a boil and add barley. Cook until tender, 50–60 minutes for hulled or hull-less barley, or 20–30 minutes for pearl barley. Drain and spread on a baking sheet to cool.
2. Heat vegetable oil in a pan over medium-high heat and fry shallots until golden brown, about 5–7 minutes. Use a slotted spoon to remove shallots and drain on paper towels, reserving the oil.
3. In a separate skillet, heat olive oil over medium-high heat until it just begins to smoke. Add mushrooms in a single layer and cook until golden on one side, about 3 minutes. Season with salt and pepper, toss, and continue cooking until fully golden, around 5 more minutes.
4. Reduce heat to medium, then add thyme, garlic, and butter to the mushrooms. Baste mushrooms in foaming butter until they emit a nutty aroma. Remove mushrooms, discarding thyme and garlic.
5. In a large bowl, mix cooled barley with cilantro, parsley, lemon juice, Parmesan, and reserved shallot oil. Season with salt and pepper. Add mushrooms and stir to combine.
6. Top the salad with fried shallots and extra shaved Parmesan before serving.