Preheat grill over medium high heat. Grill corn for 15-20 minutes, turning every 5 minutes. Meanwhile, brush sliced squash, zucchini, and asparagus with oil and season generously with Kosher salt. Grill vegetables for approximately 5 minutes, turning once, until just cooked through. Remove all vegetables from grill and allow to cool slightly. Slice asparagus into pieces. Remove husk and silk from corn and slice kernels off the cob.
Cook pasta according to package directions in a large pot and drain. Add pasta back into the pot and add grilled vegetables and arugula pesto. Toss to coat. Divide pasta between four bowls and top generously with grated parmesan cheese.
for the Arugula Pesto
Place all ingredients in a food processor and combine thoroughly. Season with Kosher salt.
My favorite parmesan cheese, by far, is parmigiano reggiano. It’s worth the price and really helps make this dish special.