Season chicken breasts with salt and pepper. Cut a long pocket into the thickest part of each breast. Slide Parma ham and sage leaves into the pockets and close tightly.
Heat oil and butter in a pan over medium-low heat. Fry the chicken breasts on both sides for 10 minutes until the internal temperature reaches 74°C (165°F). Remove and keep warm.
Clean the mushrooms and quarter them. Fry them in the same pan over medium heat for 5 minutes. Remove and set aside.
Add 250 ml of water and the chicken broth cube to the pan and bring to a boil over medium-high heat.
Stir in the dry sherry, fresh parsley, and cream.
Add the mushrooms back to the pan and heat for 2 minutes over medium heat.
Return the chicken breasts to the pan on top of the sauce and serve with mashed potatoes or pasta.