Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain and reserve about 1 cup of the pasta water. Place the pasta and 1 cup of cheese in a large bowl. Toss until coated.
In a medium skillet, heat the oil over medium-high heat. Add the mushrooms, garlic, 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook, stirring frequently, for 5 to 6 minutes until the mushrooms begin to brown. Add the tuna, olives, tomato paste, capers, red pepper flakes, 1 1/2 teaspoons salt and 1/4 teaspoon pepper. Using a wooden spoon, break up the tuna into 1-inch pieces. Add the chicken broth and bring the mixture to a boil. Remove the garlic and discard.
Pour the tuna mixture over the pasta, add the remaining 1/2 cup cheese and basil. Toss until coated, using the reserved pasta water to loosen the sauce. Season with salt and pepper, to taste, and serve.