Prepare the grill for direct cooking over high heat (450° to 550°F).
Bring a large pot of salted water to a boil (for the pasta).
In a large skillet over medium heat, warm 1 tablespoon of the oil. Add the tomatoes and cook until slightly softened, 3 to 4 minutes, stirring occasionally. Remove from the heat and add the thinly sliced basil and ¼ teaspoon of the salt.
Combine the remaining 1 tablespoon oil and the garlic, brush the mixture on the salmon, and season evenly with the remaining ¼ teaspoon salt and the pepper.
Grill the salmon, skin side up, over direct high heat, with the lid closed, until you can lift the fillets off the cooking grates without sticking, 6 to 7 minutes. Turn and continue cooking to your desired doneness, 2 to 3 minutes more for medium rare. Remove from the grill and tent with foil to keep warm.
Meanwhile, cook the pasta according to package directions. Drain the pasta and reserve ½ cup of the pasta water. Transfer the pasta immediately to the skillet with the tomato-basil mixture. Add additional pasta water, about 1 tablespoon at a time, as needed to coat the pasta.
Flake the salmon into bite-sized pieces and discard the skin. Gently fold the salmon into the pasta and garnish with basil. Serve warm.
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