Heat a gas grill to medium or prepare a medium-hot charcoal fire. Trim the pork of any excess fat and silverskin (see How to trim a tenderloin) . Rub the pork with 1 Tbs. of the oil, the oregano, 1/2 tsp. salt, and 1/4 tsp. pepper. Set the pork on the hottest part of the grill and close the lid. Grill, covered, turning once, until an instant-read thermometer inserted into the center of the pork registers 140° to 145°F, about 15 to 20 minutes. Transfer the pork to a clean cutting board and let rest for 5 minutes.
While the pork is on the grill, combine the remaining 1/3 cup oil, the parsley, walnuts, garlic, capers, lemon juice, mustard, and anchovy in a blender. Pulse the mixture until it forms a coarse paste, about five or six pulses. (This salsa verde may be made a day ahead, covered, and refrigerated. Bring it to room temperature before serving.)
Slice the pork thinly and serve with the salsa verde spooned over it.
Serve with a salad of Bibb lettuce and avocado.