Season the pork chops on both sides with salt and pepper. Heat 1 teaspoon of olive oil in a skillet over medium-high heat, and pan fry the chops until they are lightly browned and no longer pink in the center, 2 to 3 minutes per side. An instant-read thermometer inserted into the center should read at least 145 degrees F (63 degrees C). Set the chops aside.
Heat 1 more teaspoon of olive oil in the skillet, and cook and stir the shallots and thyme until the shallots start to become translucent, about 1 minute. Pour in the port wine, blackberry juice, chicken broth, and balsamic vinegar. Bring the mixture to a boil, scraping off and dissolving any brown flavor bits from the skillet into the sauce. Cook until reduced by a third, about 5 minutes. Mix cornstarch and water into a paste, and stir into the sauce. Cook until thickened, stirring constantly, about 1 minute. Reduce heat to low, and stir in the blackberries. Simmer until berries are hot.
Return the chops to the skillet, and turn to coat with sauce. Serve hot, topped with sauce.