Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the mushrooms and cook, undisturbed, until they start browning, about 2 minutes. Stir and continue cooking, stirring occasionally, until tender, 4 to 5 more minutes. Push the mushrooms to one side of the skillet. Add another 1 tablespoon olive oil and the garlic to the other side of the skillet and cook, stirring occasionally, until softened, about 30 seconds. Stir the garlic into the mushrooms and season with salt and pepper. Let cool.
Meanwhile, combine the parsley, chopped thyme and rosemary, 3 tablespoons olive oil and a pinch each of salt and pepper in a small bowl; set aside.
Butterfly the pork: Position your knife about one-third of the way up the length of the pork loin, with your knife parallel to the cutting board. Begin cutting into the pork, pulling the meat away with your other hand so that it opens up into a flat, evenly thick piece. Season the pork with salt and pepper, spread the herb oil over the top, and top with the mushrooms. Reroll the pork into a log and tie in 1-inch intervals using kitchen twine. Season with salt and pepper.
Set a rack in a large roasting pan. Heat the remaining 2 tablespoons olive oil in a large skillet over medium-high heat. Add the pork, fat-side down, and cook until browned, about 3 minutes. Continue to cook, turning, until browned all over, 1 to 2 more minutes. Transfer the pork, seam-side down, to the rack in the pan. Roast on the lower oven rack until a thermometer inserted into the thickest part of the meat (not the filling) registers 135 degrees F, 1 hour to 1 hour 15 minutes. Remove the pork on the rack to a cutting board, tent with foil and let rest 30 minutes.
Meanwhile, skim off any excess fat from the juices in the roasting pan and add the sherry. Place the pan across two burners over medium-high heat. Add the thyme and rosemary sprigs and cook, scraping up any browned bits with a wooden spoon and stirring, until the liquid is reduced by about half, 1 minute. Pour into a small saucepan, add the chicken broth and bring to a boil. Lower the heat to a simmer and cook until slightly reduced, 4 to 5 minutes. Stir in the butter and season with salt. Strain the sauce through a fine-mesh sieve.
Place the pork on a cutting board and remove the twine; cut into 1/2-inch-thick slices. Transfer to a platter and serve with the sauce.