How to Make It
Preheat oven to 450°F. Heat oil in a large skillet over medium. Add onions, 1 tablespoon thyme, and 1/4 teaspoon salt; cook, stirring occasionally, until onions are lightly browned, about 15 minutes. Add wine; cook, stirring occasionally, until liquid evaporates, about 2 minutes. Spread onion mixture in a rimmed baking sheet.
Combine butter, pepper, remaining 1 tablespoon thyme, and remaining 1 1/2 teaspoons salt in a bowl. Loosen skin on chickens; spread butter mixture under skin. Place chickens, skin sides up, on onion mixture.
Roast chickens at 450°F until a thermometer inserted in thickest portion registers 165°F, about 35 minutes. Let stand 10 minutes. Remove and discard skin; slice meat. Using a slotted spoon, reserve 1/2 cup onion mixture. Shred meat from 1 chicken; store shredded chicken and remaining onion mixture in separate airtight containers in refrigerator up to 5 days. Serve remaining sliced chicken with reserved 1/2 cup onion mixture.