Gather the ingredients.

Preheat an outdoor grill for medium-high heat and lightly oil the grate.

Season chicken breasts with oregano, basil, salt, and pepper.

Cook on the preheated grill until no longer pink in the center and the juices run clear, 5 to 6 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Remove from the grill and keep warm.

While the chicken is cooking, heat butter and oil in a saucepan over medium heat until butter melts. Add garlic and sauté until fragrant, about 2 minutes. Whisk in flour until incorporated, about 1 minute.

Whisk in cream and broth; cook, stirring constantly until mixture thickens, 3 to 5 minutes. Add Parmesan and cream cheese; cook, stirring occasionally, until both have melted, about 5 minutes.

Spoon sauce over chicken breasts and serve remaining sauce on the side.