Peel, halve, and thinly slice the onions. In a large soup pot, heat 2 tablespoons of the olive oil over medium-low heat and saute the onions gently, with a sprinkle of salt, for 30-90 minutes. The onions will soften and turn golden brown. Stir them occasionally and don't try to hurry the process.
In a medium saucepan, combine the barley with 4 cups water and a teaspoon of Kosher salt. Bring to a boil, lower the heat, cover the pan and simmer for 30-40 minutes until cooked.
Meanwhile, de-core, quarter, and thinly slice (or julienne) the cabbage. If using, trim the green onions and slice the green and white parts into ½-inch pieces; you should have about 1 cup. Chop the fresh parsley. Set aside in a medium bowl.
Wipe dirt off the mushrooms with a damp cloth, cut off their dry stem ends, and thinly slice the mushrooms. Cut very large mushrooms in half.
In a large saute pan, heat the remaining 1 tablespoon olive oil and the 2 teaspoons butter over medium-high heat. Add the minced garlic and stir for a minute, then add the mushrooms, pinch or two of Kosher salt, and three grinds black pepper. Lower the heat to medium and saute until the mushrooms turn golden brown, about 15-20 minutes.
Deglaze the mushroom pan by add the fresh thyme and the sherry. Stir until the sherry cooks away.
When the mushrooms are nicely browned and the pan is deglazed with sherry, add them to the onions in the soup pot, along with the vegetable broth. Use a little of the broth to further deglaze the mushroom pan, swirling it around to loosen any brown bits. Add the cabbage, green onions, parsley, and paprika. Let the soup simmer, covered, for a half-hour, until the vegetables are tender.
If serving immediately, stir in the cooked barley and the water from the barley pot. Otherwise, stir in the water but reserve the cooked barley in another container.
Stir in the fresh lemon juice and add more salt and pepper as deserved. The lemon juice acts to balance the sweetness of the onions, while the paprika brings in a slight spiciness.