Prep the skillet: Lightly coat a large cast iron skillet with avocado oil.
Build the base: Add the cauliflower florets and shallot rounds to the skillet to form an even layer. Place the lemon halves cut-side down among the vegetables.
Season the vegetables: Season the cauliflower, shallots, and lemon with kosher salt, fresh ground pepper, and half of the dill. Reserve the remaining dill for the chicken.
Add the sauce base: Pour the heavy cream and chicken broth into the skillet around the vegetables.
Season the chicken: Pat the chicken skin completely dry. Place the chicken on top of the vegetables, drizzle with avocado oil, and season generously with salt, pepper, and the remaining dill.
Roast: Roast in a 400°F oven until the chicken reaches an internal temperature of 165°F, about 1½ hours.
Rest: Remove from the oven and let rest for 15 minutes before carving.