Rinse the red onion in cold water, breaking up the rings and removing any white membrane between the layers. Drain well. Toss the red onions, vinegar, 1 1/2 cups water, 2 teaspoons salt and the sugar in a medium bowl until the sugar and salt dissolve. Cover with plastic wrap and let sit at room temperature until ready to serve.
Heat 2 tablespoons of the oil in a large heavy pot or Dutch oven over medium-high heat. Cook the pork, turning once, until golden brown on two sides, 4 to 5 minutes. Transfer to a plate.
Add the yellow onions to the same pot, season with salt and cook, stirring occasionally, until lightly browned, 3 to 4 minutes. Add the garlic, tomato paste, cumin and oregano and cook, stirring occasionally, until the paste is brick red and the mixture is fragrant, about 1 minute. Add the ancho and pasilla chiles, bay leaves, raisins, broth and 1 tablespoon salt. Return the pork and any accumulated juices to the pot. Bring to a boil, reduce to low, cover and simmer until the pork is falling apart and easily shreds, 1 1/2 to 2 hours.
Remove the bay leaves and discard. Transfer the pork to a medium bowl. Reserve the cooking liquid.
Let the hot cooking liquid cool for 5 minutes or so, then transfer it to a blender, filling no more than halfway. Remove the center cap from the lid and place it on the blender. Cover the lid with a folded kitchen towel to catch splatters, and pulse until smooth. Pour the sauce into a medium bowl and repeat with any remaining cooking liquid. Season with salt.
Pour 1/2 cup of the chile sauce over the pork. Use a potato masher to smash the pork into the sauce; it should fall apart completely. Remove any remaining large pieces of fat and discard. Stir to combine and season with salt if necessary.
Arrange a rack in the center of the oven and preheat to 425 degrees F.
Heat the remaining 1 cup oil in a medium skillet over medium-high heat until it bubbles immediately when the edge of a tortilla touches the surface. Working one at a time, fry the tortillas until just starting to brown and crisp, about 10 seconds per side (they should still be somewhat pliable). Drain on paper towels.
Dip both sides of each tortilla in the chile sauce just to coat and transfer to a rimmed baking sheet. Spread 1/2 cup of the chile sauce in a 6-inch strip down the center of a 13-by-9-inch baking dish.
Spoon 1/4 cup shredded pork across the center of a tortilla. Fold one side over the filling, then roll up the tortilla. Place seam-side down in the prepared baking dish. Repeat with more sauce and remaining tortillas (the enchiladas should be nestled right up against each other in the pan). Top with any remaining sauce in your bowl or leftover on the baking sheet. Cover tightly with foil and bake until the sauce bubbles, 20 to 25 minutes. Remove the foil, top with the cheese and bake until the cheese is melted but not brown, 5 to 10 minutes. Let sit 10 minutes.
Top the enchiladas with the pickled red onions. Serve with lime wedges for squeezing over.
Cook’s Note Dried ancho and pasilla are available nationwide in most grocery stores. Look for chiles that are soft and pliable, not hard and brittle—a sign they are old and have little flavor. Quesillo is a white, semi-soft Mexican-style cheese, also known as queso Oaxaca, or asadero. Mozzarella makes a fine substitute. 