In a medium bowl, combine 2 tablespoons olive oil, parsley, lemon juice, salt, and pepper.
Add the chicken breasts and coat evenly. Let stand at room temperature for about 20 minutes.
Preheat an outdoor grill or grill pan over medium-high heat.
Remove the chicken from the marinade and discard the remaining marinade.
Grill the chicken for 3–4 minutes per side until grill marks develop.
Reduce the heat, cover, and continue cooking until the chicken reaches an internal temperature of 165°F (74°C).
Transfer the chicken to a cutting board and let it rest for 5 minutes before slicing.
While the chicken cooks, drizzle the remaining olive oil over the cut sides of the sourdough and grill until lightly toasted.
Slice each bread half into serving portions.
Assemble the Chicken Caprese Sandwich by layering sliced grilled chicken, fresh mozzarella, tomato slices, basil leaves, and a drizzle of balsamic glaze.
Finish with a little extra olive oil, salt, and freshly ground black pepper before serving.