Cook the pasta according to the package directions. Reserve 1 cup of the cooking water; drain the pasta and return it to the pot.
Meanwhile, heat grill or grill pan to medium-high. In a large bowl, toss the onions, bell peppers, oil, and ½ teaspoon each salt and pepper.
Grill the vegetables, turning occasionally, until tender, 8 to 10 minutes. Transfer to a cutting board and cut into bite-size pieces.
Add the grilled vegetables, spinach, reserved pasta water, and ¾ cup of the Parmesan to the pasta and toss to combine.
Top with the basil and the remaining ¼ cup Parmesan before serving.