Bring a large pot of lightly salted water to a boil. Place noodles in the pot, cook 8 to 10 minutes, until al dente, and drain.
Melt the butter in a large skillet over medium heat. Stir in the shallots, red bell pepper, yellow bell pepper, and garlic. Cook 5 minutes, until tender but crisp. Remove vegetables from skillet, and set aside.
Place the chicken in the skillet. Season with tarragon, salt, and pepper. Cook 10 minutes, or until juices run clear.
Return the vegetables to the skillet with the chicken. Mix in the half and half, Monterey Jack cheese, and Parmesan cheese. Continue cooking 5 minutes, until cheese is melted. Serve over the cooked noodles.
