Sprinkle a thin layer of salt onto both sides of the halved chicken, about 2 teaspoons per half. Leave overnight in the fridge for the dry brine to take effect.
Preheat a smoker to 300 degrees F (150 degrees C). Toss in a couple of apple wood chunks to get some smoke.
Sprinkle the chicken with some rub. Place the chicken, diced potatoes, and shallots in the foil pan.
Place the foil pan and lemon halves in the smoker. Smoke for about 1 hour or until an internal temperature of 165 degrees F (75 degrees C) is reached.
Remove the chicken with a slotted spoon and keep warm while you finish the sauce.
Pour the liquid from the foil pan into a saucepan. If you don’t have enough liquid, use some chicken stock to create about 1 1/2 cups (12 ounces|750ml) of liquid. Add the thyme and simmer the liquid until it is reduced by half.
Add a heaping tablespoon of grated Parmesan cheese. Stir with a whisk until the cheese has dissolved. Whisk in the cold butter. Tip: Cut the butter into small pieces and add a bit at a time so the sauce thickens.
Just before you serve, add some juice from a smoked lemon and the capers. Taste and adjust the amount of lemon to your liking. Some people like it really lemony others mild. Season to taste with salt and pepper.
Cut the chicken and arrange on a plate. Add the potatoes and shallots. Pour the sauce over the chicken. Garnish with a sprig of thyme and serve.