Pork Volcánes al Pastor

1 (2-lb.) piece skinless, boneless pork shoulder (Boston butt)
7 guajillo chiles, seeds removed
3 morita chiles, seeds removed
4 garlic cloves
1/3 cup fresh orange juice
¼ cup fresh lime juice
3 Tbsp. achiote (annatto) paste
1 Tbsp. Diamond Crystal or 1½ tsp. Morton kosher salt
8 (6"-diameter) corn tortillas
4 Tbsp. extra-virgin olive oil
12 oz. quesillo (Oaxaca cheese), coarsely grated
Chopped white onion, coarsely chopped cilantro, and lime wedges (for serving)

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 ?" 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 a rack in middle of oven; preheat to 350°F. Evenly space out tortillas on a baking sheet and bake until lightly toasted and very crisp (edges will curl and pucker), 35–45 minutes.
Heat 1 Tbsp. oil in a large nonstick skillet over high until smoking. Spread about one-fourth 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°F. 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.