Preheat the broiler. Put the mushroom caps gill-side up on a baking sheet. Broil until they release their juices, about 7 minutes. Flip the mushrooms, brush the tops with 1 tablespoon vegetable oil and broil until lightly browned and the juices have cooked off, 7 to 10 more minutes. Let cool slightly.
Meanwhile, thinly slice the cabbage and onion. Heat 2 tablespoons vegetable oil in a large nonstick skillet over medium-high heat. Add the cabbage, onion, caraway seeds and ½ teaspoon salt and cook, stirring, until the cabbage is crisp-tender, about 7 minutes. Add the vinegar and 2 tablespoons water and cook until reduced, 7 more minutes. Stir in the parsley; season with salt and pepper.
While the cabbage cooks, heat about 1/2 inch of vegetable oil in a large deep skillet over medium-high heat. Lightly beat the eggs with 1/2 teaspoon salt in a shallow bowl. Put the flour and breadcrumbs in separate shallow bowls. Press the mushrooms with paper towels to flatten and absorb any liquid. Coat in the flour, then the egg, then the breadcrumbs. Fry in batches until browned and crisp, 1 to 2 minutes per side. Drain on paper towels and sprinkle with salt. Serve with the cabbage and applesauce.