Preheat a grill to medium-high. Shred the tomatoes on the large holes of a box grater set in a bowl. Stir in the garlic and 1 tbsp. EVOO; season with salt and pepper. Add the basil sprigs and let sit at room temperature for at least 20 minutes.
In a large pot of boiling, salted water, cook the pasta until al dente; drain, reserving 1/2 cup of the pasta cooking water. Return the pasta to the pot.
While the pasta is cooking, brush the eggplant with a little EVOO and season with salt and pepper. Lightly oil the grill grate and grill the eggplant, covered and turning halfway through, until grill marks appear and the flesh is soft, about 10 minutes. Season the chicken with salt and pepper and grill, covered and turning halfway through, until nicely marked, about 10 minutes. Set aside to rest for 10 minutes.
Chop the eggplant and chicken and add to the pasta. Add the torn basil leaves and the onion. Remove the basil sprigs from the sauce and discard. Toss the sauce with the pasta, adding some of the reserved pasta cooking water to thin the consistency, if desired; season with salt and pepper.