Season and Sear the Beef. Pat the chuck roast dry, then season with salt and pepper. Heat oil in a skillet and sear the roast on all sides until deeply browned. This step adds serious flavor, so do not skip it if you can help it.
Build the Sauce. In the same pan, sauté chopped onion until softened. Add garlic and cook briefly. Stir in tomato paste, chili powder, cumin, smoked paprika, and oregano. Cook for about a minute to wake up the spices. Add beef broth and scrape up the browned bits.
Slow Cooker: Add beef and sauce to the slow cooker. Cook on low 8 to 10 hours or high 5 to 6 hours.Instant Pot: Add beef and sauce to the pot. Cook on high pressure 60 to 70 minutes, then natural release for 15 minutes.Oven: Place beef and sauce in a covered Dutch oven. Bake at 300°F for 3 to 4 hours, until fork tender.
Shred and Sauce. Transfer beef to a board and shred with two forks. Return the shredded meat to the cooking liquid and stir. Add lime juice and taste for salt.
Serve. Warm tortillas and fill with shredded beef. Top with onion, cilantro, salsa, and a squeeze of lime.