Heat 3 tablespoons of olive oil in a 12-inch cast iron skillet over medium heat until shimmering. Add the tortellini in an even layer. Cook, tossing occasionally, until nicely browned in spots, 2 to 4 minutes.
Add the water, kosher salt, and black pepper. Stir with a wooden spoon to combine, scraping up browned bits from the bottom of the pan. Cover with a lid or baking sheet and cook, stirring and scraping up any browned bits from the bottom of the pan occasionally, until the pasta is tender and glossy and almost all of the water has evaporated, 3 to 5 minutes.
Remove from the heat and stir in the greens, handfuls at a time, until just wilted, about 2 minutes. Stir in the grated cheese and the remaining 2 tablespoons of olive oil. Serve garnished with additional grated cheese and freshly ground black pepper.
