Drain the tuna fillets from their oil and place them in a skillet with a drizzle of extra virgin olive oil.
Flake the tuna fillets using a wooden spoon and sauté until lightly browned.
Meanwhile, prepare the chili peppers, basil, garlic, soaked capers, and rinsed anchovies.
Debone the anchovies and finely mince the garlic, basil, capers, and anchovies together. Place the minced mixture into a small bowl along with the black olives.
Add the chopped chili peppers to the browned tuna, then pour in the tomato purée along with the minced herb and anchovy mixture.
Simmer the sauce over medium-low heat, adding thyme to taste if desired.
Cook the tagliolini in a large pot of salted boiling water until al dente. Drain and toss directly in the skillet with the sauce to coat thoroughly. Sprinkle with chopped parsley and serve immediately.
