Season chicken with salt and pepper and lightly coat with flour.
Put water on to boil for pasta. Meanwhile, in a large skillet over medium high heat, add olive oil. Add chicken and cook until golden brown. Remove chicken from skillet; tent to keep warm. Reduce heat to medium. In the same skillet, melt the butter. Add the leeks and mushrooms and saute until softened. Add vinegar and reduce by three-fourths. Add cream and tomatoes and cook until thickened. Add half the basil and all the cheese; reduce heat to keep warm.
Cook pasta according to package directions. Add pasta to the cream sauce and toss with sauce. Garnish with the remaining basil. Serve chicken breasts alongside pasta.
Note: if you can't find tarragon vinegar, make your own by combining 1 quart champagne vinegar, 3 whole peppercorns and 1/4 cup tarragon and letting it rest for 24 hours. If using dried pasta rather than fresh, start cooking the pasta before making the sauce so it comes together at the same time.