Bring a pot of salted water to boil. Add pasta and cook until al dente. Drain water and run pasta under cold water to stop cooking. Set aside.
Meanwhile, heat 2 tablespoons olive oil in a heavy-bottomed skillet over medium heat. Season salmon filets with salt and pepper. Add salmon, skin side down, to pan and cook for about 6 minutes. Flip and cook on other side for 1-2 minutes longer, until salmon is cooked through. Transfer fish to plate and cover with foil while preparing agretti.
In the same pan, add an additional tablespoon of olive oil. Add garlic and tomatoes and cook until garlic becomes fragrant, about 3 minutes. Toss is agretti and cook until it softens, about 5 minutes. Turn off heat and stir in lemon juice and zest. Add pasta to pan and mix well. Stir in Parmesan cheese and toss to coat. Adjust salt and pepper to taste. Serve salmon over pasta.