Step 1In a heavy saucepan, combine the water, barley, oats and salt. Bring to a boil over medium-high heat, then remove from heat, cover and set aside several hours, up to overnight, to soften the grains.
Step 2Return the barley mixture to a gentle simmer and continue to cook until it is very tender and the porridge thickens slightly, 10 to 20 minutes.
Step 3Meanwhile, heat 2 tablespoons olive oil in a heavy, medium-size skillet over medium-high heat. Add the mushrooms, 2 tablespoons shallot, 2 tablespoons chopped parsley, 1 teaspoon thyme and the garlic to the skillet and sauté until the mushrooms are browned and tender, about 10 minutes. Transfer the mushrooms to a bowl and set aside in a warm place.
Step 4Add the remaining olive oil, shallot, chopped parsley and thyme to the same skillet and sauté over medium-high heat until tender, about 1 minute. Add the broth and bring to a boil, scraping up any browned bits from the bottom of the pan.
Step 5Poach eggs: Pour enough water into a skillet to reach depth of 1 inch. Bring water in the skillet to a gentle simmer over medium-low heat. Add 1 tablespoon vinegar to skillet. Carefully crack each egg into individual ramekins. Gently slide 1 egg at a time into the simmering water. Cook until whites are just set, about 3 minutes. Using slotted spoon, remove eggs from the water and drain well.
Step 6Divide hot porridge between two bowls and gently spoon over the mushrooms. Place an egg next to the mushrooms and pour over the broth. Sprinkle with the cheese and freshly ground black pepper. Garnish with chopped parsley and serve.