Preheat the broiler. Cover the potatoes with water in a pot; season with salt, cover and boil until the potatoes are fork-tender, 15 minutes.
Meanwhile, heat the oil in a stovetop casserole dish or shallow enamel pot over medium-high heat. Add the onion, carrots, celery, turnips and garlic. Season with salt and pepper and cook until the vegetables brown, 8 minutes. Add 1 1/2 cups cooking liquid from the potatoes to the casserole dish. Lower the heat and scrape up any browned bits with a wooden spoon. Tie the parsley stems with twine and add to the casserole. Cover and simmer until the vegetables are tender, 8 minutes. Stir in the Worcestershire sauce, 2 tablespoons butter and the chopped veggie burgers and warm through, 5 minutes. Remove the parsley stems and stir in the chopped parsley. Keep warm.
Drain the potatoes and mash with the remaining 3 tablespoons butter and the milk; season with salt and pepper and spoon over the casserole. Sprinkle with parmesan, if desired. Broil until golden brown, 5 minutes.