Bring a large pot of water to a boil. While the water is coming to a boil, preheat your grill. Toss the broccolini with 2 tablespoons olive oil and 1/4 teaspoon salt until evenly coated. You might need to use your hands to massage the oil into the broccolini.
Spray grill with non-stick cooking spray and add the broccolini and corn directly to the grill. Close cover and cook for about 3-5 minutes, tossing occasionally and turning corn every couple of minutes. Broccolini should be slightly charred and tender, but still crisp. The corn should not be cooked all the way through, but should begin to char on the outside.
Let veggies cool while you cook the pasta. Add 1 tablespoon salt to the boiling water, then add the pasta. Cook according to package. When pasta is done cooking drain and rinse under cool water for 10 seconds then transfer to a big bowl.
Chop the broccolini into small pieces and cut the corn off the cob. Add the chopped broccolini, corn and halved tomatoes to the pasta. Now make the dressing.
Whisk together the garlic, lemon juice, mustard, salt and olive oil until well combined. Pour dressing over the pasta and toss to coat. Add the parmesan and fresh basil and stir. Serve at room temperature.