01 - Preheat oven to 400°F. Lightly grease a 9x13-inch baking dish or line with parchment paper.
02 - In a medium mixing bowl, combine the softened cream cheese, mashed blackberries, diced jalapeño, shredded mozzarella, minced garlic, salt, and pepper. Stir until smooth and evenly incorporated.
03 - Pat the chicken breasts dry with paper towels. Using a sharp knife, carefully slice a horizontal pocket into the thickest part of each breast, stopping about 1/2 inch from the opposite edge to keep it intact.
04 - Divide the blackberry-jalapeño cheese filling evenly among the four chicken breasts, spooning it into each pocket. Secure the openings with toothpicks if needed to keep the filling contained.
05 - Brush both sides of each stuffed chicken breast with olive oil. Evenly sprinkle with smoked paprika, dried thyme, and additional salt and pepper to taste.
06 - Arrange the seasoned chicken breasts in the prepared baking dish. Bake for 28 to 32 minutes, until the internal temperature reaches 165°F at the thickest part and the filling is bubbly and melted.
07 - Remove from the oven and let the chicken rest for 3 to 5 minutes. Carefully remove any toothpicks before serving.