1. In a mortar, pound the garlic to a very coarse paste with a pestle. Add the chopped basil in 2 batches and pound to a coarse paste. Stir in 3/4 cup of the olive oil. Season the pesto with salt and pepper, then transfer all but 3 tablespoons to a large bowl. Add the yellow and red cherry tomatoes to the bowl. Season with salt and pepper and let stand for 15 minutes.
2. Meanwhile, light a grill. Brush the eggplant with the remaining 2 tablespoons of olive oil and season lightly with salt. Grill the eggplant over a medium-hot fire, turning occasionally, until tender and browned, 8 to 10 minutes. Brush the eggplant with 2 tablespoons of the reserved pesto and grill for 30 seconds longer per side. Transfer to a plate and cover loosely with foil.
3. Brush the ricotta salata with the remaining tablespoon of reserved pesto and grill until the cheese is lightly charred, about 30 seconds per side. Add to the plate with the eggplant.
4. In a large pot of boiling salted water, cook the spaghettini until al dente. Reserve 2 tablespoons of the cooking water; drain the pasta and transfer to a large bowl. Add the pesto-marinated tomatoes and the reserved cooking water and toss well. Season with salt and pepper. Arrange the eggplant on plates and top with the spaghettini. Garnish with the ricotta salata and serve.