Preheat the oven to 400ºF. In a mixing bowl, toss the tomatoes with 1 tablespoon oil and a pinch of salt and pepper. Spread onto a parchment-lined baking sheet and roast for about 15 minutes or until soft. Set aside.
Heat a grill to medium-high. Brush the chicken with 1 tablespoon oil and season generously with salt and pepper. Grill for 5-6 minutes per side or until cooked through and the chicken registers 165ºF on a meat thermometer. Transfer to a plate, cover, and set aside.
Bring a large pot of salted water to boil. Add the spaghetti and cook according to package directions.
While the pasta cooks, prepare the pesto. Combine the basil, pine nuts, and garlic in a food processor. Pulse to combine. Add 4 tablespoons oil and 1 tablespoon water. Puree until smooth. Add the cheese and pulse to combine.
Slice the grilled chicken. Drain the pasta, reserving a cup of the cooking water. Add the pasta to a large serving dish and toss with the pesto and a bit of the cooking water, as needed, to create a sauce that coats the pasta. Add the chicken and roasted tomatoes. Toss gently to combine. Garnish with Parmesan cheese and serve.