In a medium bowl, combine the shredded chicken, basil pesto, sun-dried tomatoes, and half of the mozzarella cheese. Mix well until all ingredients are combined. Season with salt and pepper to taste.
Heat a non-stick skillet over medium heat and add 1 tablespoon of olive oil.
Place two slices of sourdough bread in the skillet.
Spoon equal amounts of the chicken and pesto mixture onto each slice of bread in the skillet.
Sprinkle the remaining mozzarella cheese on top of the chicken mixture, and then cover with the remaining slices of sourdough bread.
Drizzle the top of the sandwiches with the remaining olive oil.
Cook for 3-4 minutes on one side until golden brown, then carefully flip and cook for another 3-4 minutes until the other side is also golden and the cheese has melted.
Remove the sandwiches from the skillet and let them cool for a minute before slicing.
Garnish with fresh basil leaves and serve while warm.