Heat the oven to 200 C / 400 F. Place each chicken breast between two sheets of plastic wrap or parchment and pound with a mallet or rolling pin to an even 1 cm thickness. Season both sides with the salt and pepper.

Stir the cream cheese, mozzarella, spinach and garlic together in a bowl until evenly combined and spreadable.

Spread about 3 tablespoons of filling over each breast, leaving a 2 cm border. Don't overfill, because extra filling leaks out and scorches in the pan. Roll up from the short end, then wrap each roll in 2 overlapping slices of prosciutto, finishing seam side down. Secure with toothpicks if needed.

Heat the olive oil in a 30 cm / 12-inch oven-safe skillet over medium-high heat until it shimmers. Add the rolls seam side down, spaced about 2 cm apart. Turn a quarter turn every 1 1/2 minutes for 6 minutes in total, until the prosciutto is golden and crisp at the edges.

Transfer the skillet to the oven and bake for 20 to 22 minutes, until an instant-read thermometer in the thickest roll reads 165 F / 74 C.

Move the rolls to a cutting board, remove the toothpicks, and rest for 5 minutes. Slice into 2 cm rounds and spoon the pan juices over the top.