In a large Dutch oven over medium-high, heat the oil until shimmering. Add the tortellini, distributing them in an even layer, and cook, uncovered and without stirring, until well browned on the bottom, 3 to 4 minutes. Stir in the salami, garlic, tomatoes, ½ teaspoon pepper and 1 cup water. Bring to a simmer, then reduce to medium, cover and cook, stirring occasionally, until the tomatoes have softened and the tortellini are tender, 4 to 5 minutes.
Remove the pot from the heat. If desired, remove and discard the garlic. Stir in the cheese and basil, then taste and season with salt and pepper. Serve drizzled with additional oil and sprinkled with additional cheese.
