Cut breasts in half and place side by side on a platter. Drizzle with the vinegar and oil. Cut basil leaves and sprinkle over the breasts; sprinkle salt and pepper. Refrigerate at least 1 hour.
Grill the breasts or bake in a 350 degree oven for 30 minutes. Set aside.
Peel and finely chop shallots. Saute in butter 1 minute. Add cream, salt and pepper to taste. Boil to reduce slightly to the consistency of a light sauce, just thick enough to coat a spoon. Add mushrooms and peas and cook another 2 minutes.
Cook pasta, drain and shake out excess water. Thinly slice chicken. Add to pasta and gently toss with sauce, along with cheese. Serve immediately.