Bring a large pot of salted water to a boil over high heat. Add pasta and cook to al dente. Drain and keep warm.
Heat extra-virgin olive oil in a large skillet over medium heat, add anchovies and stir into oil until melted. Add the garlic, turn heat to low and saute for a couple of minutes. Stir in the tomatoes crushing them up, then stir in stock. Season with a little black pepper, add capers and olives and simmer a few minutes.
While sauce simmers, cut tuna into chunks and grind in food processor into coarse mixture. Pulse in the egg, bread crumbs and parsley and season with a little salt and black pepper. Remove tuna mixture from the processor to a bowl and add a drizzle of olive oil. Stir to combine. (Adding the oil will prevent mixture from sticking to your hands as you roll the balls.) Roll the tuna into small balls and drop into the sauce. Cover the pot and simmer the tuna balls in the sauce, about 4 to 5 minutes.
Place the pasta into a serving bowl. Add the tuna and sauce, toss to coat and garnish with shredded basil.