Season and sear the chicken: Season chicken with salt, pepper, and paprika. Heat olive oil in a large skillet over medium-high heat. Sear chicken until golden brown, about 3–4 minutes per side. Transfer to a plate and set aside.
Prepare the glaze: In the same skillet, add 1 tablespoon oil, garlic and jalapeño; sauté for about 1 minute until fragrant. Stir in blackberry preserves, water, apple juice, balsamic vinegar, and Dijon mustard. Bring to a gentle simmer.
Incorporate blackberries: Add half of the fresh blackberries, lightly mashing some to release their juices. Simmer briefly to combine flavors.
Finish the chicken: Return chicken to the skillet, spooning glaze over the top. Reduce heat to medium-low and simmer for 5 minutes, or until the chicken is cooked through.
Finalize the sauce: Stir in remaining blackberries. Simmer for 1–2 minutes until the sauce is glossy and slightly thickened.
Serve: Transfer chicken to a serving dish, spoon glaze over the top, and garnish with fresh basil or thyme. Optional: Finish with a small squeeze of fresh lime juice before serving for extra brightness.