Toast the chiles. Heat a dry skillet over medium heat. Add the ancho and guajillo chiles and toast for about 20–30 seconds per side, just until fragrant. Do not let them burn, as they can become bitter.
Soak the chiles. Transfer the toasted chiles to a bowl and cover with 1 cup of the beef broth. Let them soak for 20 minutes, until softened. Discard the soaking liquid.
Blend the sauce. Transfer the softened chiles to a blender. Add the onion, garlic, cumin, Mexican oregano, salt, black pepper, and 1 cup of beef broth. Blend until completely smooth.
Strain the sauce. Pour the sauce through a fine-mesh sieve into a bowl, pressing on the solids with a spoon to extract as much sauce as possible. Discard the remaining solids.
Brown the Beef
Prepare the beef. Pat the beef dry with paper towels and season it with salt and black pepper.
Brown the beef. Heat the olive oil in a large Dutch oven over medium-high heat. Working in batches, brown the beef on all sides, about 2–3 minutes per side. Do not overcrowd the pot. Transfer the browned beef to a plate.
Combine and Braise
Start the braise. Reduce the heat to medium. Pour the strained chile sauce into the Dutch oven, scraping up any browned bits from the bottom. Add the remaining ½ cup beef broth and stir to combine.
Return the beef. Return the browned beef and any accumulated juices to the pot. Add the bay leaves. Bring the mixture to a gentle simmer.
Braise the beef. Cover and reduce the heat to low. Simmer for 1½–2 hours, stirring occasionally, until the beef is completely tender. If the sauce becomes too thick before the beef is tender, add a small splash of beef broth and continue cooking.
Reduce the sauce. Remove the lid and simmer uncovered for 10–15 minutes, until the sauce has thickened enough to coat the beef.
Let it rest. Remove the chili from the heat and let it rest for 15–20 minutes.
Finish and serve. Discard the bay leaves and stir in the fresh lime juice. Taste and adjust the salt if needed. Sprinkle with fresh cilantro and serve warm.
Notes
Do not over-toast the dried chiles. Toast them for only 20–30 seconds per side, just until fragrant. If they turn dark or black, they can make the sauce bitter.
Brown the beef well. Work in batches and give the beef enough space to develop a deep brown crust. This adds a lot of flavor to the finished dish.
Season the beef before browning, but don't over-salt the sauce. The broth and sauce will reduce during cooking, concentrating the salt.
Let the chili rest for 15–20 minutes before serving. This lets the sauce settle and the beef absorb more chile flavor.