Heat the oil in a large saucepan, over medium-high heat. Add the leeks and cook, stirring occasionally, until softened, 3-4 minutes. Add the mushrooms and cook until leeks and mushrooms are very tender, about 5-6 minutes.
Stir in the garlic and oregano, and cook 2 minutes.
Pour in the wine and cook, stirring, until evaporated, about 1-2 minutes. Add the kale and cook until just wilted, about 1 minute.
Stir in the barley and cook, stirring, 30 seconds. Pour in 3 cups of the broth and bring to a boil, reduce heat to medium, and cover. Simmer until the barley is almost tender, about 30 minutes.
Add the remaining broth. Uncover and simmer until the barley is tender. 12-15 minutes longer.
Remove from the heat and stir in the parsley, cheese, salt, and pepper.