Prepare the marinade by combining 1/4 cup balsamic vinegar, 2 tablespoons olive oil, 1 cup crushed strawberries (about 1 cup from the pound), 1 teaspoon kosher salt, and 1/2 teaspoon black pepper in a bowl.
Place the 4 chicken breasts in a zip-top bag or shallow dish and pour the marinade over them. Seal or cover and refrigerate for 2–6 hours.
When ready to cook, heat a cast-iron or large skillet over medium-high and add the remaining 2 tablespoons olive oil.
Remove the chicken from the marinade, discarding the used marinade, and sear the breasts 2–3 minutes per side until golden brown. Transfer the seared chicken to a plate and set aside.
Add the thinly sliced 1/2 red onion to the skillet and sauté 1–2 minutes until beginning to soften.
Add the remaining halved strawberries, the remaining 1/4 cup balsamic vinegar, and the remaining salt and pepper to the skillet. Simmer 5–6 minutes, stirring occasionally, until the strawberries break down and the sauce thickens.
Nestle the seared chicken and any juices back into the skillet among the berries. Reduce heat to medium-low, cover, and simmer gently 10–15 minutes until the chicken reaches 165°F, flipping the chicken a few times and stirring the sauce to prevent sticking.
Serve each chicken breast topped with sliced fresh mozzarella and torn basil leaves, with extra strawberries if desired.