One or two days before serving the chicken, spatchcock the chicken by removing the backbone and wingtips with kitchen shears or a sharp knife. Save these to make chicken stock.
Lay the bird flat and push down on the breastbone to further flatten it out. Salt generously on both sides and leave to sit, in room temperature, for 30 minutes.
In a small bowl, whisk together 2 tablespoons kosher salt with the buttermilk until dissolved. Pour into a large, resealable bag and add the thyme, black peppercorns and chicken. Close and squeeze the bag to thoroughly coat the chicken. Place the bag on a rimmed sheet pan and refrigerate overnight or up to two days.
Remove chicken from refrigerator one hour before cooking. Preheat oven to 425 degrees.
Cut each fennel bulb into six or eight pieces. Toss chopped fennel, potatoes and lemon slices on a rimmed baking sheet or tray with 2 tablespoons olive oil, salt and pepper. Set aside.
Remove chicken from marinade and wipe off any excess buttermilk and herbs with paper towels. Arrange the chicken in a cast iron skillet or large baking pan. Season the bird lightly with pepper and some additional salt, if desired. Place vegetables and chicken in the oven.
Roast for 20 minutes. Lower heat to 400 degrees, rotate the pans and roast for another 25 minutes or so, until the juices run clear when a knife is inserted between the leg and thigh.
Rest chicken for 10 minutes on cutting board. Cut into pieces at the joints and split the 2 breasts into 4 pieces. Serve immediately with the roasted vegetables, garnishing with fennel fronds.