Preheat the oven to 350°F. Place a cooling rack on top of a rimmed baking sheet.
Lightly season the chicken breasts with salt and pepper on both sides.
Add 3 tablespoons of the butter to a large sauté pan, over medium-high heat. Pan-sear the chicken breasts, browning on both sides. Remove the chicken from pan. (Chicken will not be fully cooked at this stage)
Cut slits lengthwise in the chicken breasts, about three-quarters of the way in, creating a pocket in which to stuff the chicken. Divide the sun-dried tomatoes and goat cheese evenly between the 4 breasts and stuff each one. After stuffing them, wrap each breast tightly with 2 pieces of prosciutto.
Place the chicken on the cooling rack and bake for 25 minutes. Baking the chicken on top of the cooling rack will allow the prosciutto to crisp up on both sides.
While the chicken is baking, add the remaining 3 tablespoons of butter and the garlic to the same pan the chicken was cooked in. Over medium heat, sauté until the butter is browned and the garlic is fragrant, about 2 minutes. Be careful not to burn the garlic as it will taste very bitter.
De-glaze the pan with chicken stock. Using a rubber spatula, scrape the pan and mix in any remaining bits of chicken. Let simmer for 5 minutes.
To the pan add the heavy cream, pesto and Parmesan cheese. Reduce the heat to low and simmer for 10 minutes. Stir frequently so the sauce does not get too thick or stick to the bottom of the pan.
Plate the chicken, top with sauce, and garnish with toasted pine nuts.