If you are serving the pork with salsa verde, make that first. The salsa verde can be made up to 3 days in advance and kept in the fridge until ready to use.
Combine the 1/2 cup chopped apricots, the 1/3 cup pistachios, the 3 sprigs of chopped rosemary, the 2 tablespoons honey, the clove of minced garlic, 1/2 teaspoon kosher salt, and 1/4 teaspoon ground black pepper in a medium bowl.
Slice pork loin halfway down its side, leaving about 1/2-inch border on the other side (don't cut all the way through) and butterfly the roast open (see video for step-by-step instructions). Using your palm press both sides down to make them as flat as possible. Using a meat mallet pound the pork until it is of even thickness and about 1 inch thick on all sides. Season both sides generously with 2 teaspoons salt and 1 teaspoon pepper.
Place the fat-side down and spread filling over pork, leaving an inch border on all sides. Starting on one end, roll pork up, trying not to let too much of the filling fall out. Tie with 3 or 4 pieces of kitchen twine. Stuff any filling that has fallen out back into the middle of the pork. They might be a lot of stuffing right at the seam, stuff it back into the middle so it doesn't burn on the grill. Make sure the fat layer is on the outside of the roast.
Prepare your grill for 2-zone grilling, creating one zone of direct medium-high heat (425°F-450°F) and a second zone of indirect heat. If using a gas grill, simply light half of its burners to medium high. If using a charcoal grill, distribute all of the lit charcoal to one side of the grill. The side of the grill directly over the heat source is the direct heat zone, while the other side is the indirect heat zone. Clean the grill grates with a grill brush.
Place the pork roast on the indirect heat side of the grill (not over the coals). Close the lid and cook. You want the temperature of the grill to be between 425-450 degrees Fahrenheit. The warm air from the direct heat side of the grill circulates throughout the grill, cooking the pork very gently – much like an oven. Every 15 minutes, carefully rotate the pork 180 degrees to promote even cooking & check the internal temperature. The pork is ready once an instant-read meat thermometer inserted in the thickest part of the pork loin registers 140-145 degrees Fahrenheit. The cooking time will vary depending on how hot your grill is and how thick the pork loin is. It will take anywhere from 45 minutes-1 hour 15 minutes.
For best results, remove the pork loin to a cutting board and tent with aluminum foil. Let it rest 15-20 minutes to allow the natural juices redistribute themselves back into the pork, then carve into thick slices and serve with the salsa verde.