A day ahead: In a large bowl, dissolve the salt in 8 cups cold water. Add the chicken and submerge it in the brine, using a plate to weight it, if needed. Refrigerate for 24 hours.
On the day of grilling: Set up a charcoal or gas grill to cook with direct heat.
For gas grills, turn all the burners to low (an oven thermometer set on the grate should read 450°F to 500°F; you’ll be able to hold your hand 4 or 5 inches above the grate for 5 seconds).
For charcoal, light a chimney starter filled with mesquite lump charcoal. When all the coals are lit, pour them in the center of the grill and spread in an even layer. Let them burn down to medium-low heat (you’ll be able to hold your hand 1 to 2 inches above the grate for 5 full seconds).
Meanwhile, heat 1 Tbs. of the oil in a 12-inch skillet over medium heat. Add the lemon slices. Cook, turning occasionally, until they’re soft and golden brown (it’s fine if the pulp starts to dissolve), 5 to 10 min. Transfer the slices to a plate and let cool.
Meanwhile, take the chicken out of the brine; pat it dry with paper towels. Cut off the wingtips at the first joint. Remove the backbone by cutting down each side of it; discard it or save it for stock. Lay the chicken flat on a work surface, skin side up, and press on the breastbone to flatten the chicken; you'll feel some of the rib bones breaking.
Use your fingers to separate the skin from the breast, thighs, and drumsticks, being careful not to tear the skin. Spread the rosemary under the skin, distributing it evenly over the breasts, thighs, and drumsticks. Working with one slice at a time, slide the lemon slices under the skin of the chicken, keeping them in a single layer. Sprinkle the skin with salt and pepper. Drizzle the remaining 1 Tbs. of oil on the chicken and rub the skin so it's completely coated.
Oil the grill grate. Set the chicken on the grill, skin side down. If using gas, cover the grill; if using charcoal, don’t. Grill the chicken until the skin is deeply golden brown, 10 to 15 minutes. Monitor the grill for hot spots and move the chicken as needed. Flip the chicken, cover the grill (keep the vents open for charcoal), and grill until an instant-read thermometer registers 175°F in the thickest part of the thigh (the breast should be at least 165°F) and the juices run clear when the thermometer is removed, another 15 to 35 minutes. Remove from the grill and let rest for 10 minutes before carving and serving.