Preheat the oven to 400°F (200°C). Pat the chicken breasts dry. In a bowl, combine the cottage cheese, mozzarella, Parmesan, chopped blackberries, jalapeño, garlic, lemon zest, and a pinch of black pepper.
Carefully cut a horizontal pocket into the thickest side of each chicken breast, stopping before cutting through the opposite edge.
Mix the smoked paprika, garlic powder, oregano, salt, and black pepper. Season the outside of each chicken breast evenly.
Divide the filling between the four pockets, leaving enough room to close them. Secure the openings with toothpicks if needed.
Heat the olive oil in a large oven-safe skillet over medium-high heat. Sear the chicken for 3–4 minutes on the first side and 2–3 minutes on the second, until golden.
Transfer the skillet to the oven, or move the chicken to a lightly greased baking dish. Bake for approximately 12–18 minutes, depending on thickness, until the thickest part of the chicken reaches 165°F (74°C).
While the chicken bakes, combine the sauce blackberries, balsamic vinegar, optional honey, water, and a small pinch of salt in a saucepan. Simmer over medium-low heat for 4–6 minutes, gently crushing some of the berries until the sauce thickens. Add a little water if necessary.
Rest the cooked chicken for 5 minutes. Remove all toothpicks, spoon the blackberry sauce over the breasts or serve it alongside them, and enjoy.