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Preheat oven to 400°. Remove stems from mushrooms and wipe the caps clean a damp paper towel. Rub caps with about ½ tablespoon of olive oil. Arrange, stem sides up, in a roasting pan, oven-safe skillet, or on a baking sheet. Bake 15 minutes, until they tenderize a bit and begin to release their juices.
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While mushrooms bake, coat the bottom of a medium skillet with 1 tablespoon of olive oil and place over medium heat. Add onion and sauté until soft and translucent, about 5 minutes. Add garlic and sauté 1 minute more. Add beans to skillet, flip a few times to incorporate, and allow to cook until heated throughout, about 2 minutes. Use a fork or potato masher to lightly mash about half of the beans. Stir in parsley, basil, breadcrumbs and red wine vinegar. Season with salt and pepper to taste.
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After portobellos have baked for 15 minutes, remove them from the oven and spread about 2 tablespoons of puttanesca sauce inside of each cap, using a spoon to spread it around a bit. Stuff each cap with a quarter of the bean mixture, pressing firmly. Spoon 3-4 tablespoons more of puttanesca sauce on top of bean mixture. Return portobello caps to the oven and bake 6-8 minutes more, until mushrooms are very tender.
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Spoon a bit more puttanesca sauce on onto the bottom half of each roll. Stuff rolls with portobellos, and top with additional puttanesca sauce, if desired. Serve.