Preheat the oven to 475 degrees F.
Liberally season the chickens, inside and out with thyme, and salt and pepper, to taste. Rub each bird with 1/2 stick of softened butter.
Cut the ends off 2 bulbs of garlic to expose all of the cloves for roasting. Put the potatoes and onions in a roasting pan, and liberally drizzle with extra-virgin olive oil and salt and pepper, to taste. Dress the whole cut bulbs of garlic with extra-virgin olive oil and season with salt and pepper, to taste. Press the bulbs into the rosemary leaves so that some adheres to the garlic, then sprinkle the remainder over the potatoes and onions. Put 1 chicken into this pan. Sprinkle with fennel seed or pollen. Squeeze the juice of 1 lemon over the chicken and put the lemon in cavity of bird along with 2 large cloves of crushed garlic. Tie the legs together with string. Add the dressed garlic heads to the pan and roast this bird for 15 minutes at 475 degrees F. Baste the bird with pan juices, then lower the temperature 375 degrees F and roast until the juices run clear, about 1 hour to 1 hour and 15 minutes.
Add the artichokes to the pan the last 20 to 30 minutes of roast time, mixing them in with the potatoes and the fennel. Remove the chicken to a cutting board. Let rest for 10 minutes and then carve. Arrange the chicken on a serving platter along with the potato mixture. Pour the pan juices over the chicken. When ready to serve, divide the roasted garlic among the plates and serve with crusty warm bread for mopping.
For the second bird, put it in a roasting pan and douse it with the juice of 2 limes, then stuff the limes into the bird along with 2 large cloves of smashed garlic. Tie the legs and roast for 15 minutes at 475 degrees F, baste, then roast 45 minutes. Remove the chicken from oven to a cutting board. When cool enough to handle, remove the chicken from bone and pull the meat into small pieces. Reserve for a make-ahead meal.