Remove the chuck roast from the refrigerator at least 30–60 minutes before cooking to allow it to come to room temperature. Pat the roast dry all over with paper towels.
In a small bowl, whisk together the olive oil and Worcestershire sauce until fully blended.
In a second small bowl, combine the garlic powder, onion powder, smoked paprika, dried thyme, dried rosemary, coarse black pepper, and kosher salt. Stir until well combined.
Using a pastry brush or your fingers, apply the olive oil and Worcestershire mixture all over the chuck roast, covering all sides thoroughly. Sprinkle the dry rub mixture generously over all surfaces and press it firmly into the meat with your hands.
Line the air fryer basket with aluminum foil, leaving the sides slightly raised to catch drippings. Preheat the air fryer to 390°F (200°C) for 5 minutes.
Place the seasoned chuck roast into the preheated air fryer basket, fat side up if possible. Cook at 390°F (200°C) for 15 minutes to sear. Flip the roast using tongs after 15 minutes.
Reduce the air fryer temperature to 320°F (160°C) and continue cooking for an additional 45–60 minutes. For medium-rare, aim for 135°F (57°C); for medium, 145°F (63°C); for medium-well, 155°F (68°C). Check the internal temperature starting at the 30-minute mark.
Once the roast reaches your target internal temperature, remove it from the air fryer and place it on a cutting board. Tent loosely with foil and let it rest for at least 10–15 minutes before slicing.
If making gravy, pour the drippings from the foil into a small saucepan over medium heat. Whisk in the flour and gradually add the beef broth, whisking constantly until thickened (3–5 minutes). Season with salt and pepper to taste.
Slice the roast against the grain into thin to medium-thin cuts (approximately ¼ inch or 6mm thick). Arrange on a serving platter, spoon gravy over the top if desired, and serve immediately.