1. Make the brine: Fill a large stockpot with 16 cups water; bring to a simmer over medium-high heat. Add sugar, salt, lemons, olives and their brine, parsley, garlic, peppercorns, thyme, and bay leaves; stir to dissolve sugar and salt. Remove from heat and let stand for 20 minutes to infuse.

2. Add ice to brine mixture to cool. If brine is not completely cold, transfer to refrigerator until cold. Add chickens to cold brine and weight down with a plate or small pot lid to keep submerged. Transfer to refrigerator and let brine 8 to 12 hours.

3. Make the chicken: Cut preserved lemons into quarters. Cut flesh away from rinds, reserving both. Remove chicken from brine; discard brine. Rinse chickens and pat dry using paper towels. Place 1 chicken on work surface with legs facing you. Starting at the cavity, work the handle of a wooden spoon between skin and one breast to create a pocket, working slowly and gently to avoid tearing skin. Repeat process with other side.

4. Holding chicken in place with one hand, slide index and middle fingers of your other hand into each pocket to enlarge it, then slide your fingers down to create a pocket over the thigh. Repeat entire process with second chicken.

5. Insert pieces of preserved lemon rind, white pith-side down, and thyme sprigs into pockets over thighs and breasts. Sprinkle cavities with salt and rub chickens with reserved flesh from preserved lemons, scraping off any pieces of lemon flesh that stick to the chicken and discard.

6. Position 1 chicken, breast side up, with legs facing you. Cut a vertical slit in one side, about 1 inch back from cavity, alongside thigh. Cross end of opposite drumstick over drumstick on this side and poke top end of drumstick through slit. Depending on condition of chicken's skin, it may rip as you try to poke the drumstick through it, so have some kitchen twine on hand just in case to tie legs together if necessary. Repeat process with other chicken. Season chickens with salt.

7. Preheat oven to 500 degrees.

8. Make the vegetables: Place turnips, carrots, onions, parsnips, and rutabagas in a large bowl; add olive oil, thyme, garlic, and 3 pinches salt. Toss to combine. Transfer vegetables to a large roasting pan spreading in an even layer. Set a roasting rack over vegetables. Place chickens, breast-side up on roasting rack, leaving space between chickens. Chickens should not be touching vegetables, but set over them.

9. Transfer roasting pan to oven; roast for 15 minutes. Spread room-temperature butter from remaining chicken ingredients over the breasts; continue roasting 30 minutes more. Roast until skin is deep golden brown and an instant-read thermometer inserted into thickest part of thigh reaches 160 degrees, 15 to 20 minutes more. Transfer chicken to a cutting board and let stand 20 minutes.

10. Meanwhile, remove rack and set roasting pan on stove over two burners. Bring cooking liquid to a simmer. Whisk in preserved lemon liquid and simmer for 2 minutes. Stir in butter, piece by piece, to emulsify sauce and glaze vegetables. Stir in parsley. Serve chickens whole or carved over vegetables.