Butterfly each chop from the fat side and trim them of excess fat. One at a time, place each chop between 2 sheets of plastic wrap and pound to an even ¼-inch thickness. Lightly brush or spray the paillards with oil and season all sides with salt and pepper.
Remove the dark green tops off of each leek, cutting about 2 inches above the point where the leaves begin to darken. Trim just enough of each root end to remove the stringy parts, but leave enough of each root end so the layers remain attached. Cut each leek in half lengthwise. Remove the tough outer leaves on each leek. Rinse the leeks under water, opening up the layers to remove any dirt. Pat dry. Lightly coat the leeks with oil and season to taste with salt and pepper.
Prepare the grill for direct cooking over low heat (250° to 350°F).
Brush the cooking grates clean. Grill the leeks over direct low heat, with the lid closed as much as possible, until softened and slightly charred on all sides, 15 to 20 minutes, turning every couple of minutes for even cooking, and moving over indirect heat if the leeks become too dark before they are tender.
In a food processor combine the sauce ingredients. Pulse until you get a semi-smooth consistency. Season to taste with salt.
Increase the temperature of the grill to high heat. Brush the cooking grates clean. Grill the paillards over direct high heat for about 3 minutes on the first side, turning when the meat is nicely marked. The second side will need only a minute to finish cooking.
Transfer the paillards, with the first grilled side facing up, to a serving platter or individual plates. Divide the sauce evenly and spoon over the meat. Arrange two leeks on top of each paillard. Garnish with some parsley over the top.