Boil water for pasta. Add 1/2 tablespoon kosher salt, add pasta, and cook for 2–5 minutes, depending on thickness.

Meanwhile, place bacon and olive oil in a medium-size skillet over medium heat, stirring occasionally. Cook bacon until crisp, about 5 minutes. Remove bacon and set aside. Pour off all except 1 tablespoon of bacon fat.

Add tomatoes to the skillet, coating with bacon fat, and season with salt and pepper. Cook over medium heat until tomatoes almost melt—about 5 minutes. Add half of the bacon and combine with tomatoes. Add pasta directly to skillet, pour in 1/8 cup pasta water, and toss to coat. Add greens and toss with pasta until wilted. Add more pasta water, a tablespoon at a time, as needed to loosen up the sauce. Season with more salt, pepper, remaining bacon, and Parmesan.