Preheat the oven to 425℉. Bring a large pot of water to a boil.

In a 9x13 baking dish, combine the tomatoes, shallot, garlic, olives, capers, 1/3 cup of the olive oil, red pepper flakes, salt, black pepper, lemon juice, balsamic vinegar, parsley, and basil. Toss until well combined. Transfer to the preheated oven and bake, uncovered, until the cherry tomatoes begin to burst, about 15 minutes.

Remove from the oven and toss. Nestle the salmon filets in the cherry tomatoes and drizzle with the remaining olive oil. Transfer back into the oven and bake until the salmon is cooked through and flakes easily with a fork, about 8 minutes.

Meanwhile, add the linguine to the boiling water and cook until al dente according to package instructions. Drain the pasta and transfer to a large serving bowl.

Flake the salmon into bite-sized pieces then carefully pour the salmon-tomato mixture over the cooked pasta and gently toss until the pasta is just coated in the sauce. It might seem a little overly runny at first, but the pasta will drink up the liquid from the cherry tomatoes within a few short minutes. Taste and add more salt to taste, if desired.

To serve, garnish with more fresh basil and crushed red pepper flakes, if desired.