Preheat the oven to 400 degrees F.
In a large bowl, stir the zucchini, yellow squash, peppers, tomatoes, chickpeas and garlic with 2 tablespoons olive oil, 1 teaspoon kosher salt, 1/2 teaspoon black pepper.
Spread the vegetables, chickpeas and garlic on 2 sheet pans in a single layer. Arrange the tomatoes cut side up.
Roast the vegetables until they are browned at the edges and the chickpeas are crispy, about 25-30 minutes. Rotate the pans in the oven halfway through roasting. Let the vegetables cool for 10 minutes.
While the vegetables are roasting, simmer the pasta in a large pot of salted boiling water until al dente according to package instructions. Drain and rinse with cold water in a colander.
For the vinaigrette, squeeze the roasted garlic cloves into a small bowl and mash and whisk them with the mustard, 1/2 teaspoon kosher salt and 1/2 teaspoon black pepper, vinegar and 1/4 cup olive oil.
Combine the roasted veggies and chickpeas in a large bowl with the pasta, scallions, chives and Parmesan. Then stir in the vinaigrette.