First, prepare the pesto chicken. In a small bowl, toss thinly sliced chicken with extra virgin olive oil and 1 tbsp of the pesto until well-coated. Sauté in a skillet over medium-high heat for about 6-8 minutes, or until cooked through.
Meanwhile, preheat a cast iron griddle over medium heat.
Next, assemble the panini. Spread about 1 tsp butter on the outside of each slice of sourdough bread, then carefully place one slice of bread butter side down on a plate. Spread the remaining pesto onto the bread, then top with mozzarella cheese slices. Layer with spinach leaves, then the cooked pesto chicken, then top with shredded mozzarella cheese. Close the sandwich with the remaining slice of bread (butter side up).
Carefully transfer the panini to the hot cast iron griddle and cook until bottom is crisped and golden-brown, about 3 to 4 minutes. Carefully flip and cook until the bottom is crisped and golden-brown again, another 3 to 4 minutes. (Alternatively, you can use a preheated grill press on top of the sandwich to crisp both sides at once and avoid flipping your panini.)
Transfer panini to a cutting board and slice diagonally in half. Serve immediately.