Gently clean mushrooms with a damp paper towel. Separate the stems from the caps of the mushrooms. Dice the mushroom stems. Cut the mushroom caps into ¼-inch slices.
Pour ¼ olive oil into a large, heavy-bottomed saucepan or dutch oven. Add the diced mushroom stems, chopped onion, and diced celery and carrots, and set the pan over medium heat.
Add the anchovy fillets and dried thyme to the pan and cook, stirring to break up the anchovies, for 1 minute longer. Pour in the chicken broth and soy sauce and turn the heat up to high.