How to Make It
In a bowl, cover the cannellini beans with water and let soak overnight. Drain the beans and transfer to a medium saucepan. Add 5 cups of water and bring to a boil. Simmer over low heat, stirring occasionally, until tender, about 1 hour. Drain the beans in a colander set over a bowl; reserve the cooking water.
Cook the fava beans in boiling salted water until tender, 3 to 5 minutes. Drain the favas in a colander set over a medium saucepan. Add the reserved cannellini bean cooking water to the fava water and stir in the mushroom stems, mint stems and shallot skins. Simmer over low heat until reduced to 2 cups, about 20 minutes; strain.
In a large skillet, heat 2 tablespoons of the olive oil. Add the mushrooms and shallots and cook over moderate heat, stirring, until the mushrooms are lightly browned, about 5 minutes. Push the mushroom mixture to the side and add the sausage balls in an even layer. Cook until browned, about 3 minutes per side. Add the chopped mint and the wine and cook until the wine evaporates. Add the cannellini and fava beans and their cooking liquid. Cook over high heat until the liquid is reduced by half, about 10 minutes.
Remove the skillet from the heat and let stand for 5 minutes. Season with salt and pepper. Spoon the sauté onto 8 plates and scatter the mint leaves over the top. Drizzle with the remaining 1/4 cup of olive oil and serve.