Step 1

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.

Step 2

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.

Step 3

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.

Step 4

Place a rack in middle of oven; preheat to 350°. Evenly space out tortillas on a baking sheet and bake until lightly toasted and very crisp (edges will curl and pucker), 35–45 minutes.

Step 5

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.

Step 6

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.