Start by spatchcocking the chicken (see instructions and video in the blog post). Season the chicken liberally all over, including the underside and under the skin. Place the chicken on a baking sheet lined with foil and transfer to the fridge uncovered for at least 4 hours, up to 48 hours.
Preheat an oven to 400 F.
Heat a large cast iron skillet (at least 11" in diameter), over medium heat. Add the neutral oil. Once the oil is hot, lay the chicken in the skillet skin side down and weigh it down with another skillet or heavy pot. Let the chicken cook for 10-12 minutes, until deeply golden brown.
Remove the skillet weighing the chicken down and gently flip the chicken over so that it's sitting in the skillet skin side up. Arrange the lemon slices around and under the chicken. Transfer to the oven and roast for 35-40 minutes, until the thickest parts of the breasts and legs register 165 F on a meat thermometer.
Remove the chicken from the oven and transfer to a cutting board to rest. Return the skillet to the stove over medium heat. Add 2 tablespoons of the cold butter to the pan and use a silicone spatula to work up any browned bits. Once the butter is melted and foamy, add the shallots and cook for 2-3 minutes until soft. Add the capers and cook for 1 minute.
Add the white wine and lemon juice to the skillet and bring to a simmer. Use a spoon to work up any more remaining brown bits from the skillet. Reduce the heat slightly and whisk in the remaining cold butter, one tablespoon at a time. Whisk constantly so that the butter emulsifies into the sauce. Add the lemon zest and fresh herbs. Season to taste with salt and pepper.
Place the chicken on a serving platter and pour the lemon caper sauce over the top. Serve immediately.