Prepare the Beef: Pat the beef chuck dry with paper towels. Season generously all over with salt. Heat 1 tablespoon of olive oil in a large heavy-bottomed pot or Dutch oven over medium-high heat until shimmering. Brown the beef in batches, turning occasionally, until a deep, crusty brown sear develops on all sides (this should take about 3-4 minutes per side). You'll hear a satisfying sizzle as the beef hits the hot pan. Remove the beef to a plate and set aside.

Sauté Aromatics and Toast Chillies: Reduce the heat to medium. Add the remaining 1 tablespoon of olive oil to the pot. Add the chopped white onions and sauté, stirring occasionally, until they soften and turn translucent, about 5-7 minutes. You should smell their sweet aroma filling your kitchen. Add the smashed garlic cloves and cook for another minute until fragrant, being careful not to burn them. Meanwhile, in a dry pan over medium heat, toast the dried ancho, guajillo, and arbol chillies for 30 seconds to 1 minute per side until they become fragrant and slightly pliable – be careful not to burn them, as this will make them bitter. You'll notice a distinct, smoky perfume.

Build the Sauce Base: Add the toasted chillies, chopped tomatoes, cumin, oregano, black pepper, cloves, and cinnamon to the pot with the onions and garlic. Stir well to combine, letting the spices bloom for about 1-2 minutes until their aroma intensifies. Pour in the apple cider vinegar and stir, scraping up any browned bits from the bottom of the pot.

Simmer the Birria: Return the browned beef to the pot. Pour in the beef stock, ensuring the beef is mostly submerged. Add the bay leaves. Bring the mixture to a gentle boil, then reduce the heat to low, cover the pot tightly, and simmer for at least 3 hours, or until the beef is incredibly tender and falls apart easily with a fork. You should see a gentle, consistent bubbling and smell the rich, complex aroma of the stew developing.

Blend the Consommé and Shred the Beef: Carefully remove the beef and bay leaves from the pot and set the beef aside to cool slightly. Strain the liquid and solids through a fine-mesh sieve into another large pot, pressing down on the solids to extract as much liquid as possible. Discard the bay leaves and any tough chilli stems. Transfer the strained solids (onions, garlic, chillies, tomatoes, spices) to a blender along with a cup or two of the strained braising liquid. Blend until completely smooth, then return this vibrant, thick sauce back into the pot with the remaining braising liquid. Taste the consommé and adjust salt if needed. Shred the cooled beef using two forks until it's in tender strands.

Assemble and Cook the Quesadillas: Heat a large non-stick frying pan or griddle over medium heat. Dip one side of a tortilla into the birria consommé, coating it lightly but thoroughly (it should be a deep, reddish-orange colour). Place the wet side down on the hot pan. Sprinkle generously with grated cheese, then add a good portion of the shredded birria beef over one half of the tortilla. Once the cheese starts to melt and the tortilla on the bottom crisps up to a golden-brown (about 2-3 minutes), fold the other half of the tortilla over the beef and cheese. Press gently with a spatula to seal. Cook for another 1-2 minutes until both sides are beautifully golden and crispy, and the cheese is fully melted and gooey. Repeat with the remaining tortillas, wiping the pan clean between batches if necessary. Serve the hot quesadillas immediately with small bowls of the warm birria consommé for dipping, garnished with fresh coriander, diced white onion, and lime wedges.