Combine the goat cheese, basil, and black pepper together in a small bowl and mix until well combined.

Using a meat mallet, pound each chicken breast into an even ½-inch thickness. Divide the cheese mixture between the chicken breasts, then fold each one around the cheese. Wrap a slice of prosciutto around the outside of each piece of chicken.

Preheat the grill to medium-high heat and lightly oil the grate. Grill the chicken for 12-15 minutes, turning occasionally, until the chicken is cooked through and the inner temperature reaches 165°F.