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An herby, punchy green sauce transforms a humble pork shoulder into a centerpiece. Serve it at your next big gathering for very happy guests.
Make the green sauce: Process garlic scapes, arugula, parsley, mint, lemon juice, and 2/3 cup oil in a food processor until a smooth sauce forms, 30 to 45 seconds. Scrape down sides of bowl. Add 1 teaspoon salt and 1/2 teaspoon pepper; process until well combined, about 15 seconds. Transfer to a liquid measuring cup or bowl.
Preheat oven and prepare pork: Preheat oven to 350°F. Place pork on a cutting board. Using a sharp chef’s knife, score fat side of pork in a crosshatch pattern, with lines about 1 inch apart and 1/4 inch deep. Season pork on all sides with remaining 2 1/2 tablespoons salt and 3 teaspoons pepper.
Sear pork: Heat a 12-inch oven-safe skillet or Dutch oven over medium-high until very hot, about 2 minutes. Add remaining 2 tablespoons oil; swirl to cover bottom of skillet. Using large tongs, carefully place pork, fat side down, in hot oil. Sear until fat crisps, 3 to 4 minutes. Sear remaining sides of pork until golden brown, 1 to 3 minutes per side; adjust heat as needed to prevent overbrowning. Remove skillet from heat. Turn pork fat side up. Slather half of green sauce over top of pork; reserve remaining green sauce for serving. Pour 1 cup water into bottom of skillet.
Roast pork: Roast for 1 hour. Baste pork with pan juices and continue to roast until an instant-read thermometer inserted in center of pork reads 165°F, another 1 hour to 1 hour, 15 minutes.
Cool, slice, and serve: Transfer pork to a cutting board and let rest for 15 minutes. Using a sharp knife, slice pork against the grain into very thin slices. Transfer sliced pork to a platter. Spoon some pan juices over pork. Serve pork with remaining green sauce.