Heat the beef broth in a saucepan, and keep it warm.
In a large, deep skillet, heat 2 tablespoons (30 ml) of the butter over medium heat. Add the shallots and garlic and cook, stirring, until starting to soften. 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 tablespoon (15 ml) of butter. Cook, stirring, for about 1 minute. Add the wine, and cook until the wine is absorbed, about 3-4 minutes.
Add 1 cup (250 ml) of the hot beef broth, and stir to combine. Cook over medium heat, stirring once or twice, until nearly all of the broth has been absorbed. *It is important not to boil the mixture, but rather to let it simmer gently. Adjust heat to that point. Continue adding ladles of broth one a time, allowing to cook until the liquid is absorbed before adding another. Cook until all the broth us added. Test the barley. It must be al dente. If it is not quite there, add another ladle of hot water and cook a bit longer. When cooked, stir in grated parmesan cheese and stir to combine.
Season dish with salt and freshly ground pepper, to taste. Serve with additional grated parmesan, and garnish with fresh thyme sprigs or fresh thyme leaves.
