In a large pot, bring about 4 cups stock plus 1 cup water to a boil. Add the barley and cook, stirring occasionally, until tender, 22 to 25 minutes.
While the barley is cooking, in a small saucepan, bring the mushrooms and about 2 cups stock to a simmer over medium. Reduce the heat to low and keep warm until the mushrooms are reconstituted, 15 to 20 minutes. Using a slotted spoon, transfer the mushrooms to a work surface; chop. Reserve the mushroom soaking liquid.
In a large skillet with a lid, heat the EVOO, three turns of the pan, over medium-high. Add the zucchini, onion and garlic; season with salt and pepper. Partially cover and cook, stirring occasionally, until the vegetables are tender, 6 to 7 minutes.
Add the vegetables, mushrooms and reserved soaking liquid (avoid any grit that may have settled in the pan) to the pot with the barley and bring to a simmer. Remove from the heat and add the lemon juice, parsley and mint; season. If soup is too thick for your liking, add more water or stock. Serve in shallow bowls and top with the cheese.