The mushrooms in this salad are tender and slightly caramelized when they come out of the oven. The barley adds texture and the radicchio adds freshness and a touch of bitterness, while the honey in the dressing helps to sweeten everything up.
Place a large rimmed baking sheet in oven: preheat to 450°F. Bring a large pot of water to a boil over medium-high heat. Stir in barley; cook, uncovered, stirring occasionally, until tender, 20 to 25 minutes.
Meanwhile, whisk vinegar, soy sauce and 2 tablespoons oil in a large bowl. Add mushrooms to the mixture and toss to coat. Carefully remove the hot pan from the oven; spread the mushrooms on it in an even layer. Roast until tender and golden, about 20 minutes.
Drain the cooked barley and transfer to a large bowl. Immediately add 1 tablespoon each oil and lemon juice and 1/4 teaspoon each salt and pepper.
Add the roasted mushrooms, radicchio, parsley and Parmesan to the barley mixture. Whisk honey and the remaining 1 tablespoon each oil and lemon juice and 1/4 teaspoon each salt and pepper in a small bowl until combined. Drizzle over the salad and toss to coat.