Heat 2 tablespoons olive oil in a large pot over medium-high heat. Add the onion, 1/2 teaspoon salt and a few grinds of pepper and cook until the onion is just tender, about 5 minutes. Add the garlic and cook 1 minute. Stir in the wine to deglaze the pot and scrape up any browned bits, then simmer until mostly evaporated, 1 to 2 minutes. Stir in the chicken broth, tomatoes, beans, 1/2 teaspoon salt and a few grinds of pepper and bring to a boil. Reduce the heat and simmer, stirring occasionally, until the tomatoes are soft and the mixture is thickened, 12 to 15 minutes.
Meanwhile, preheat the broiler. Toss the sausages with the remaining 2 tablespoons olive oil on a baking sheet. Broil, turning halfway through, until well browned and cooked through, 6 to 8 minutes. Let rest about 5 minutes.
Remove the bean mixture from the heat and stir in the arugula until just wilted; spoon into bowls. Slice the sausages in half and add to the bowls. Top with basil and cheese.