Tbsp. plus ½ cup extra-virgin olive oil, divided, plus more for grill
skin-on, bone-in chicken breasts (about 3 lb.)
Kosher salt, freshly ground pepper
garlic cloves, crushed
large sprigs thyme
cup torn pitted Castelvetrano olives
Prepare a grill for 2-zone heat (for a gas grill, set one burner to medium-high and one or two burners to low; for a charcoal grill, bank most of the coals on one side). Lightly oil grate. Pat chicken breasts dry; season generously all over with salt and pepper. Place on a rimmed baking sheet; let sit at room temperature at least 30 minutes and up to 1 hour.
Thinly slice 1 lemon crosswise into rounds; pluck out seeds. Place half of lemon slices in a small bowl. Add garlic and 1 Tbsp. oil and toss to coat; season with salt and pepper. Set remaining lemon slices aside for serving. Slice remaining lemon in half and squeeze juice into another small bowl (you should have about ¼ cup). Set aside.
Pat chicken dry again (the salt will have drawn out more moisture) and rub with 2 Tbsp. oil. Grill oiled lemon slices over hot side of grill, turning once, until charred in spots, about 3 minutes. Transfer to a plate. Grill chicken on cooler side of grill, skin side down, until skin is browned and starting to crisp, 15–20 minutes. Turn chicken over, cover, and continue to grill until an instant-read thermometer inserted into the thickest part of breasts registers 160°, 8–10 minutes longer. If desired, uncover grill, move chicken over to hotter side, and lightly char a minute or two. Transfer to a cutting board and let rest 10–15 minutes.
Pull chicken meat from bones and slice ½" thick. Place on a rimmed platter, shingling slices. Scatter thyme sprigs, olives, grilled lemon slices, and reserved fresh lemon slices over. Season with more salt and pepper; drizzle reserved lemon juice and remaining ½ cup oil over. Let sit at least 15 minutes and up to 1 hour before serving.