Bring a large pot of salted water to a boil over high heat. Preheat a gas grill to medium-high (or prepare the charcoals for a charcoal grill).
Add the peas to the boiling water and cook until tender and bright green, about 2 minutes. Using a small mesh sieve or slotted spoon, retrieve the peas from the boiling water and transfer to the bowl of a food processor (reserve the pot of boiling water). Add the mint, garlic, cheese, walnuts, lemon zest and juice to the bowl with the peas. Pulse the machine a few times to chop the ingredients up and then, with the machine running, stream the oil into the bowl to make a paste. Season with salt and pepper and reserve.
Add the pasta to the boiling water and cook to al dente according to package directions. Reserve about 1/2 cup of the pasta cooking water and then drain the cooked pasta and reserve.
While the pasta is cooking, drizzle the vegetables with the olive oil and season with the garlic powder, salt and pepper. Grill the vegetables, turning occasionally, until blackened and tender, 4 to 5 minutes. Let the cooked vegetables cool enough to handle, cut the kernels off the corn, and then coarsely chop all the vegetables together.
In a large serving bowl, toss the pesto, cooked pasta and grilled vegetables together. Add a few splashes of the reserved pasta cooking water as needed to thin out the pesto and evenly coat everything. Serve the pasta garnished with additional grated cheese.