In a large pot, cook the onion, stirring in 1/4 cup water. Stir in the carrot, potato, 1 cup water, tomato paste, thyme, and bay leaves. Bring to a boil and simmer briskly untill the potato and carrot are cooked, about 20 minutes. (Add extra water if necessary to keep the vegetables covered.) Near the end of the cooking time, add the vegan wine, beans, onion, garlic and salt. Return to a boil, lower the heat and simmer uncovered, 10 minutes more. Remove the bay leaves.
Meanwhile, spray a small skillet with nonstick cooking spray, and cook the mushrooms, stirring over low heat. Combine with the beans and serve.