For the spice rub: Toast the fennel seeds, coriander seeds, and peppercorns in a small heavy pan over medium heat. When the fennel turns light brown, add the cinnamon, salt, chili powder, and red pepper flakes and toss. Immediately turn the spice mixture out onto a plate to cool. Put into a food processor with the small bowl attachment and blend until the spices are evenly ground. Makes about 1 cup.
For the brine: Combine all the ingredients, except ice, in a stockpot, bring to a boil, and stir to dissolve the solids. Remove from the heat, add ice, and cool to room temperature. Refrigerate until thoroughly chilled.
Put the chickens in the brine, cover, and refrigerate for 4 to 8 hours. Turn the chickens once half way through brining. Remove the chickens from the brine and pat them dry. Remove the backbones and split the chickens in half. Coat with 1/2 cup toasted spice rub on all sides, pressing into the skin. Put the chickens on a cookie sheet and put them into the refrigerator, uncovered, overnight to dry.
Preheat the oven to 400 degrees F.
Heat a 14 to 16-inch saute pan or roasting pan over medium-high heat and add the butter. When it is melted place the chicken halves in the pan, skin side down. Cook on the stove for approximately 5 minutes, basting with the pan juices. Put chickens into the oven and roast for 20 minutes. Remove from the oven and turn the chickens over on a cookie sheet to rest for 10 to 15 minutes. Transfer chicken to a platter.
Put the roasting pan back on the stove, heat the pan juices to simmering, add the thyme and vinegar, and reduce by half. Spoon some of the vinaigrette over the arugula. Serve with tomato slices and Olive Oil Braised Potatoes. Spoon the rest of the sauce over the chicken and dig in.