Preheat a gas grill to medium-high (400°F to 450°F) on 1 side. Drizzle zucchini, yellow squash, onion, and bell peppers evenly with 2 tablespoons olive oil; and 1 1/2 teaspoons salt. Arrange sliced onions on oiled grill grates over lit side of grill. Grill covered until onion slices are slightly charred on both sides, about 3 minutes per side. Transfer to unlit side of grill. Grill covered until softened, about 8 minutes.
Add bell peppers, zucchini, and eggplant to lit side of grill; grill, covered, until slightly charred and softened, about 3 minutes per side, flipping vegetables halfway through cooking time. Let cool for 10 minutes.
Bring a large pot of salted water to a boil over high. Cook pasta according to package directions. Meanwhile, whisk together cider vinegar, mustard, honey, garlic, remaining 2 tablespoons olive oil, and remaining 1 teaspoon salt in a large bowl until well combined and emulsified, about 30 seconds; set dressing aside.
Chop grilled vegetables into about 1-inch pieces. Add to dressing in bowl along with cooked pasta and basil; toss to coat. Season to taste with additional salt, and garnish with additional basil.