Place pork shoulder on a parchment-lined rimmed baking sheet and freeze, uncovered, until almost completely frozen, about 2 hours. Using a very sharp knife, thinly slice pork (shoot for about ⅛" thick). Transfer to a large bowl.
Meanwhile, bring guajillo and morita chiles and 1 cup water to a boil in a medium saucepan. Cover, remove from heat, and let sit 30 minutes to allow chiles to soften.
Transfer chiles and soaking liquid to a blender. Add garlic, orange juice, lime juice, achiote paste, and salt; purée until smooth. Pour over pork and toss to coat. Cover and let sit at room temperature 2 hours, or chill up to 6 hours.
Place tortillas on a sheet pan in a cold oven. Set a timer for 15 minutes and heat oven to 425°. When the timer goes off, remove tortillas from the oven.
Heat 1 Tbsp oil in a large nonstick skillet over high until smoking. Spread about ¼ of pork across skillet in a single layer; cook, undisturbed, until browned underneath, about 2 minutes. Turn over and cook, undisturbed, until browned underneath and cooked through, about 2 minutes. Transfer to a platter and wipe out skillet. Working in 3 batches, repeat process with remaining pork and 3 Tbsp oil.
Increase oven temperature to 500°. Divide pork among tostadas and top with quesillo; bake until cheese is melted, 8–10 minutes. Top with onion and cilantro and serve with lime wedges.
Tags: BonAppetit, Bon-Appetit, Bon Appetit, Cheese, Dinner, Gluten-free, Main, Mexican, Nut-free, Pepper, Pork, RickMartinez, Rick-Martinez, Rick Martinez, Taco, Tortilla, Tostada
https://www.bonappetit.com/recipe/pork-volcanes-al-pastor