Stab the potato with a fork 4-5 times. Place on a microwave-safe plate and nuke for 7-8 minutes, flipping it once halfway through. Let cool. Peel and mash.
In a food processor, add the olives, peppers, basil, capers, anchovies, garlic, red pepper flakes (if using) and ½ teaspoon black pepper. Pulse 4-5 times until it's a chunky paste.
In a medium mixing bowl, combine ⅔ cup of the mashed potato, 1 cup of the olive mixture, the red wine vinegar and ½ teaspoon salt. Stir until blended. Taste and add more salt if necessary. It might already taste really salty and that's okay! It'll get dispersed evenly through the pork loin. Set the mixture aside. (No food processor? You can also chop everything somewhat finely by hand and stir together.)
Heat the oven to 375°.
Cut six 12"-long pieces of kitchen twine and set aside next to where you'll be working with the pork.
Place the pork loin on the cutting board with one of the short sides closest to you.
Flip the loin over so the fat cap is facing down. Slice down the length of the loin about ½" in from the right side (if you're right-handed).
Now turn the knife so the blade is flat and parallel with the cutting board, about ½" up from the board. Slice down the length of the loin, making sure your knife doesn't angle up or down.
Keep slicing down the length of loin. Pull the pork open as you cut. Stop about ½" from the other long side. Lay the loin open like a book and repeat the process until you can open up the loin a second time.
Cover the loin with plastic wrap and use the flat side of a meat mallet to pound any thicker areas so the loin is an even thickness.
Flip the loin over so the fat cap is facing up. Season the whole surface with salt, pepper and the Italian seasoning.
Flip the loin over again so the fat cap is facing down. Season the whole surface with salt.
Using a small offset spatula, a regular spatula or your hands, spread the olive and potato mixture evenly across the surface of the loin, stopping about 1" from the edges.
Roll up the loin from the long side on the left. Don't press too hard so you don't squeeze the filling out the sides (that being said, if some naturally falls out, that's fine).
Slide the pieces of string under the pork, evenly spreading them out. Tie each piece firmly (but again, not super tight) and trim any excess string.
Heat the oil in a large oven-safe skillet or small, stovetop-safe roasting pan over medium-high heat.
Place the loin in the pan fat cap-side down. Sear the fat until golden brown, about 5-6 minutes.
Flip the loin so the fat cap is facing up. Place in the oven for 25-30 minutes, until a thermometer inserted into the meat reads 145°. Make sure the thermometer tip isn't actually in the filling (stab the pork in a few different places to make sure).
Remove from the oven and place the loin on a cutting board. Cover loosely with foil and let rest for 10 minutes.
Scoop any big bits or chunks of fat or the olive mixture out of the pan and discard. Heat the pan over medium heat. Add the balsamic vinegar and stir to scrape any remaining bits off the pan. Stir in the butter or ghee one tablespoon at a time until you have a thick, glossy sauce. Taste and add more salt and/or pepper if necessary. Pour into a small pitcher.
Cut the string and remove from the pork. Slice and carefully arrange on a serving platter. Drizzle first with any remaining juices on the cutting board and then the balsamic sauce. Serve with any remaining sauce on the side.