Preheat a grill to medium heat (about 350°F).
While the grill is preheating bring a large pot of water to a boil. Once boiling and add the spaghetti. Cook for 11 minutes. Drain the pasta, reserving ½ cup of the pasta water.
Add the walnuts to a non-stick pan over medium low heat. Toast for 5-6 minutes. Shaking periodically to ensure an even toast. Remove from heat and let cool.
Add the tomatoes, basil and 1 tablespoon olive oil to a small mixing bowl. Toss to coat. Place a veggie grilling pan or basket on your grill and add the tomatoes. Cook tomatoes for 6 minutes. A few may “pop” as they cook. When done, remove from grill and let cool slightly.
To make the pesto add the toasted walnuts to the bowl of a food processor fitted with a standard blade and pulse 4-5 times. Add the garlic and pulse a few more times. Add red pepper flakes, parmesan and parsley. Pulse until well combined. With the motor running, add the remaining olive oil and continue to process until fully blended.
To prepare the dish, add the pasta to a large sauté pan over medium heat. Toss in the pesto and reserved pasta water, stirring to combine. Add the roasted tomatoes and heat for 1-2 minutes. Divide among four serving dishes and top with mozzarella and chopped Italian parsley. Serve immediately.