Bring a large pot of water to a boil.
Heat a grill or grill pan to medium-high.
In a large bowl, toss the onions, bell peppers, oil, 1/2 teaspoon salt, and 1/2 teaspoon pepper. Grill, turning occasionally, until tender, 8 to 10 minutes. Transfer to a cutting board and cut into bite-size pieces.
Cook the pasta according to the package directions. Reserve 1 cup of the cooking water, drain the pasta, and return the pasta to the pot. Add the grilled vegetables, spinach, the reserved pasta water, and 3/4 cup of the Parmesan to the pasta and toss to combine. Top with the basil and the remaining Parmesan before serving.