Heat the vegetable stock in a saucepan and keep warm.

In a large, deep skillet, heat 2 tablespoons of the olive oil over medium heat. Add the shallots and garlic and cook, stirring, until soft and translucent. Add the mushrooms and thyme leaves and season with salt and pepper. Cook, stirring regularly until mushrooms are tender and lightly golden, about 5-6 minutes. Add the barley and the remaining 1 Tbsp olive oil. Cook, stirring, for about 1 minute. Add the wine and cook until the wine is absorbed, about 3-4 minutes.

Add one ladle of the vegetable stock and stir to combine. Cook over medium heat, stirring once or twice, until nearly all of the broth has been absorbed. *You don't want to boil the mixture too vigorously, but you do want a consistent simmer. Adjust heat to that point. Continue adding broth, one ladle at a time, allowing it to cook until the liquid is absorbed before adding another ladle. Cook until all the broth is added. Test the barley. You want it to be al dente. If it is not quite there, add another ladle of hot water (or broth, if you have more available) and cook a bit longer. When barley is cooked, stir in grated Parmesan cheese and stir to combine.

Season the dish with salt and freshly ground pepper, to taste. Serve with additional grated Kefalotyri and garnish with fresh thyme sprigs or leaves.