Place each chicken, breast side down, on a cutting board. Using poultry shears, cut along each side of the backbone and remove it. Open each chicken like a book. Split each chicken in half by cutting right along one side of the breastbone. Coat each chicken half thoroughly with the paste. Place the chicken halves in two large, resealable plastic bags. Press the air out of the bags, seal tightly, and refrigerate for 8 to 24 hours.
Prepare the grill for indirect cooking over high heat (450° to 550°F).
Lightly brush the chicken halves with oil and season evenly with salt. Brush the cooking grates clean. Place the chicken halves, skin side down, over indirect high heat. Lightly coat the bottom of a sheet pan with oil. Wrap three bricks in aluminum foil. Place the sheet pan on top of the chicken halves and weight the pan down with the bricks. Close the lid and grill the chicken until the skin is crispy and the juices run clear, 30 to 40 minutes. Wearing insulated barbecue mitts, carefully remove the bricks and sheet pan. Turn over the chicken halves and move over direct heat for 2 to 3 minutes to crisp the underside. Transfer from the grill to a platter and let rest for 3 to 5 minutes. Cut each chicken half crosswise into two pieces. Serve warm.