Pre-heat your oven to 375 degrees. Get out a medium cast iron skillet and set it aside.
With a large sharp knife, carefully cut each chicken breast horizontally to create a pocket, being careful not to slice all the way through the breasts. Open the chicken breasts and season the insides with salt and pepper.
Spread about 1 – 1 1/2 tablespoons of the chutney onto one side of each of the open chicken breasts. Then top with the spinach and mozzarella cheese, dividing them evenly between the two breasts. Fold the top half of the chicken breasts over to close the pockets. Thread a couple toothpicks through the chicken breasts near the openings to help secure them. If some of the filling has spilled out, just tuck it back in. Season the outside of the chicken breasts with salt and pepper.
Heat your cast iron skillet over medium-high heat and warm the olive oil. Add the chicken breasts and cook for 4 minutes, or until well browned on one side. Then using a pair of tongs, carefully flip the chicken breasts and cook for an additional 4 minutes on the second side. Transfer the cast iron skillet to your pre-heated oven and bake the chicken breasts for about 20 minutes, or until the internal temperature reaches 80 degrees Celcius, flipping the breasts once during this time. Remove from the oven.
Transfer the chicken breasts to serving plates. Carefully remove and discard the toothpicks. Serve immediately.