To a medium bowl, add the blackberry preserves, water, bourbon, balsamic vinegar, olive oil, Dijon mustard, garlic, salt, and pepper. Whisk until fully combined. Set aside.
Pat the chicken breasts dry with paper towels.
In a small bowl, mix the salt, pepper, and paprika. Season both sides of the chicken breasts with the seasoning mixture.
To a large nonstick skillet over medium heat, add the olive oil. Once the oil is hot, add the chicken breasts. Cook for 6-7 minutes on the first side. Then, flip and cook for another 6-7 minutes, or until the chicken reaches an internal temperature of 165°F. Adjust the heat as needed to prevent burning. Remove the cooked chicken from the skillet and transfer to a plate. Cover loosely with aluminum foil to keep warm.
To the same skillet, add 1 cup of the fresh blackberries. (Reserve ½ cup of blackberries for serving.) Cook over medium heat for 2-3 minutes, stirring gently, until the blackberries begin to soften and release their juices.
Pour the prepared blackberry glaze into the skillet with the blackberries. Stir gently to coat the berries. Then, let the glaze simmer for 1-2 minutes to warm up.
Return the chicken breasts to the skillet. Spoon the blackberry glaze over the chicken, letting it simmer for 1-2 minutes to warm the chicken through and coat it with the sauce.
To serve, spoon the blackberry glaze over the chicken and garnish with the reserved fresh blackberries and fresh thyme.