Heat 2 tablespoons of the olive oil in a 4 qt Dutch oven or soup pot (that has a lid) over medium until shimmering. Add the leeks, onions, celery and carrots, and cook until the onions are soft, about five minutes.
While the vegetables simmer, take about ⅓ of the sliced mushrooms and give them a rough chop. Add the chopped mushrooms to the pot, and cover. Cook until the mushrooms have released their juices and are nicely browned, 8 to 10 minutes.
Remove the lid and add a glug of wine to the pot. Stir well, and let the alcohol cook off for about 5 minutes.
Scoot everything to one side of the pot. Add the remaining tablespoon of olive oil to the cleared spot. Spoon the tomato paste, smoked paprika, marjoram, and a large pinch of salt over the oil. Stir to bloom the spices and create a fragrant paste.
Add a splash of the broth to the paste, stirring to loosen into a sauce. Mix the sauce into the vegetables.
Add the remaining broth to the pot soy sauce (or Worcestershire sauce). Pour in the uncooked barley and give everything a good stir. Add the sliced mushrooms, bay leaves and thyme stems. Increase heat to bring the soup to a boil, then reduce to maintain a steady simmer while partially covered.
Cook for 45 minutes, stirring now and then to keep things from sticking to the bottom of the pot.
Begin checking the soup during the final 15 minutes of cooking. The soup will have seemed very liquidy at the start of the cook, but the barley will absorb a great deal of the liquid while it cooks. If the soup is still liquidy, remove the lid while the soup finishes.
Taste and add more salt if necessary. Finish with the chopped parsley and serve immediately.