Step 1

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

Step 2

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

Step 3

Cook the chicken breasts on 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).

Step 4

Melt butter with vegetable oil in a saucepan over medium heat. Cook and stir garlic in the butter mixture until fragrant, about 2 minutes. Stir flour through the butter mixture; cook and stir until flour is incorporated, about 1 minute more. Pour heavy cream and chicken broth into the saucepan; stir. Continue cooking until the mixture thickens, 3 to 5 minutes. Add Parmesan cheese and cream cheese to the saucepan; cook, stirring occasionally, until the cheese is melted into the sauce, about 5 minutes.

Step 5

Plate the chicken breasts. Spoon sauce over the chicken. Serve extra sauce on the side.