Place the potatoes in a large, heavy pot and fill with enough salted water to cover by 1 inch. Bring to a boil, then lower the heat and simmer for 15 minutes; drain in a colander.
Using the same pot, heat the olive oil over medium heat. Add the sausage and cook, stirring, just until no longer pink, about 8 minutes. Stir in the beans, meat sauce, cooked potatoes, chicken broth and herbes de Provence; season with salt and pepper. Bring to a simmer and cook until slightly thickened, about 20 minutes.