Preheat a gas or charcoal grill on medium high heat.
To make the pesto, finely chop the garlic, basil, mint leaves, walnuts, parmesan, 1/4 teaspoon kosher salt, 1/4 teaspoon black pepper and lemon juice in a food processor. With the motor running, pour the olive oil through the feed tube until it's smooth and combined.
Drizzle the corn and zucchini with 1 tablespoon olive oil, 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper.
Grill the corn until it is lightly charred all over, about 5-7 minutes. Grill the zucchini until it has grill marks, about 2-3 minutes per side.
When the vegetables are cool enough to handle, slice the kernels off the ears of corn and dice the zucchini.
Cook the pasta in salted boiling water until al dente according to package instructions and then drain.
Toss the pasta with half the pesto.
Fold the grilled corn and zucchini, chives, mozzarella and the remaining pesto into the pasta.