Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius).

Place chicken breasts in a single layer in a 9x13 inch baking dish.

In a medium saucepan over low heat, melt the butter and cream cheese together until smooth.

Whisk in the condensed cream of chicken soup, chicken broth, and the dry Italian dressing mix until fully combined.

Pour the sauce mixture over the chicken breasts in the baking dish.

Bake in the preheated oven for 45 minutes, or until chicken reaches an internal temperature of 165 degrees Fahrenheit (74 degrees Celsius).

While the chicken is baking, boil a large pot of salted water and cook angel hair pasta according to package directions until al dente; drain.

Serve the chicken and sauce generously over the prepared angel hair pasta.