Cover the roast with the dry rub, covering all sides.
In Dutch Oven over medium-high heat, warm olive oil until it shimmers.
Add roast and allow to sear for a minute or two. Use tongs to turn roast over and sear the other side, another minute or so. Sear all sides.
Pour Dr. Pepper over the roast.
Place lid on Dutch Oven and bake for 3 hours (roughly 1 hour per pound).
Remove from oven, take off string (if your pork shoulder was tied), and shred the meat. Place back in oven, uncovered for 30 minutes.
Notes
This recipe calls for 3 pounds of boneless pork shoulder. If your cut is larger than 3 pounds, it will need to cook longer. Typically, you would add another hour per pound. Cook to an internal temperature of 205F for shredding pork shoulder.
You can substitute the Dr. Pepper with cola if desired.