Preheat oven to 350 degrees F (180 degrees C). On a baking pan, place Walnut (1/2 cup) in oven for 6-8 minutes, or until toasted.
While walnuts are toasting, remove the root from Leek (4) with a chef's knife, and cut the remaining portion in half. Slice leeks into 1/2 inch pieces. Place in a small bowl, and submerge in water. Agitate leeks to remove dirt.
Drain liquid off, and reserve leeks. Once walnuts are toasted, remove form oven and set aside.
In a large saute pan over medium-high heat, add Unsalted Butter (1 tablespoon) and cook until the aroma is nutty, and the butter looks slightly amber in color. Remove from heat, and place cooked butter in a small kitchen bowl.
Using the same saute pan, add leeks, Unsalted Butter (1 tablespoon), Water (1 cup), Kosher Salt (to taste), and Ground Black Pepper (to taste). Place over medium-high heat and cook for 15-20 minutes, or until all the liquid evaporates. Leeks should be tender once the water evaporates.
While the leeks are cooking, prepare the Pearl Barley (1 cup). Heat a small-medium saucepot over medium heat. Add Unsalted Butter (1 tablespoon), then add pearl barley. Cook for 1 minute, stirring to ensure the grains toast evenly.
Add Water (2 1/2 cup), Kosher Salt (to taste), and Ground Black Pepper (to taste), and turn heat to high. Bring water to a boil, and lower heat to medium-low. Cover, and cook until tender, about 25-30 minutes. Remove from heat.
Add mixed Wild Mushrooms (8 ounce) and Fresh Thyme (2 sprig) to the pan with the leeks, and continue to cook over medium-high heat 4-5 more minutes while stirring. Whisk Sherry Vinegar (1 tablespoon), Lemon (1) juice, and lemon zest in the bowl of melted butter.
Once the mixed mushrooms and leeks have cooked, over medium-high heat, add cooked barley and lemon-butter mixture to pan. Cook 2-3 minutes, while stirring, to thoroughly incorporate. Remove pan from heat, discard fresh thyme.
Season with Kosher Salt (to taste) and Ground Black Pepper (to taste). Divide barley onto two plates, and top with walnuts and Parmesan Cheese (2 ounce). Enjoy!