Trim the pork tenderloin of any excess fat and silver skin.
Whisk together the lime juice, garlic powder, paprika, and salt in a bowl. Rub the pork tenderloin with the marinade in a covered container. Set aside in the refrigerator for 20 minutes or up to 24 hours.
Lightly grease a grill or grill pan with nonstick spray. Preheat the grill or grill pan on medium heat. Cook the pork for 12 to 15 minutes, turning every 2 minutes until the tenderloin reaches an internal temperature of 140 degrees F.
Remove the tenderloin from the grill and set it aside to rest for 5 minutes before slicing into thin strips.
For the sauce: add all ingredients to a blender or food processor and blend until smooth. Adjust the seasoning with salt and pepper to taste.
Prepare the Bao buns accoring to the directions on the package or if making your own see recipe in menu.
Slice the bao buns at the crease (but not all the way through) Stuff them with pork, carrot, cilantro, and cucumber. Drizzle with sauce. Serve with kimchi.