Spring Borshch With Peas and Wild Mushrooms

Name: Spring Borshch With Peas and Wild Mushrooms
Ingredients: 1½ oz. (¾ cup) dried wild mushrooms, such as morels, chanterelles, or black trumpets; 2 medium onions, 1 whole, peeled, 1 finely chopped; 1 bay leaf; 1 medium beet, peeled and cut into ⅛-in. slices; Kosher salt; 3 Tbsp. extra-virgin olive oil; 1 medium carrot, coarsely grated; 1 medium red bell pepper, sliced into thin batons; 2 Tbsp. tomato paste; 8 oz. small new potatoes, scrubbed and quartered; 2 cups shelled fresh or frozen fava beans; ¼ cup dill fronds, plus more for serving; 2 garlic cloves; 8 oz. green cabbage (about ½ head), shredded; 4 oz. shelled green peas; Beet kvass, or fruity vinegar; Crusty bread, for serving