Heat the olive oil in a large Dutch oven over medium-high heat. Add the sausage and cook, breaking it up with a wooden spoon, until browned and cooked through, 5 to 7 minutes.
Add the onion, season with salt and pepper, and sauté until softened and translucent, 3 to 5 minutes. Add the garlic and sauté until fragrant, about 1 minute. Pour in the chicken broth along with the tomatoes and their juices. Bring to a boil, then lower the heat to a simmer.
Stir in the rice and continue to simmer gently, covered but with the lid slightly ajar, stirring occasionally, until the rice is tender, 35 to 40 minutes.
Taste and season with additional salt and pepper, as needed. Stir in the basil and serve garnished with shaved Parmesan.
