Use paper towels to pat dry the chicken, then season generously with
salt and pepper on both sides. Leave on the counter for 30 to 45 minutes while you gather and prep the rest of the ingredients.

Melt the butter with the oil in a large braiser or Dutch oven over medium-high heat.

Add the chicken skin-side-down in batches, being careful not to crowd the pan (crowding the pan creates steam and will hinder browning). Let the chicken cook undisturbed for 5 minutes. Use tongs to flip the chicken over carefully. Continue cooking for another 5 minutes. Transfer the browned chicken to another pot or bowl with a tight-fitting lid and proceed with the next batch, repeating the steps until you’ve browned all the chicken.

Add all of the garlic to the pot where you browned the chicken, lower
the heat and saute for 5 to 10 minutes, until the garlic is golden. Keep
a close watch on the garlic, stirring often to keep it from burning.

Add the wine, turn up the heat and bring it to a boil while using a
wooden spoon or rubber spatula to scrape up any brown bits stuck to
the pan (these brown bits, called fond, add a ton of flavor). Lower the
heat, and return the chicken and any juices to the pot. Distribute the
thyme leaves evenly over the chicken, cover, and simmer on low for
20 minutes.

If you’re using a mix of white and dark meat, while the chicken is
simmering, wash and dry the reserve pot. After the 20-minute simmer,
check on the temperature of the white meat. The meat should be at
160 degrees or slightly higher. Transfer the white meat to the reserve
pot and cover tightly.

Continue simmering the dark meat for 10 minutes or until the internal temperature is 175 degrees (dark meat becomes more luscious cooked to a slightly higher temperature than white meat), transfer it to to the reserve pot and keep it covered (if not using white meat, simmer the dark meat undisturbed for 30 minutes on low or until it reaches 175 degrees).

Whisk together 1/2 cup of the sauce with the flour in a small bowl
until all lumps have dissolved. Pour the flour mixture into the sauce
and whisk well to thicken the sauce; simmer for 5 minutes. Raise the
heat to medium and whisk in the brown sugar and cream. Taste the
sauce, adding salt and pepper, if needed. Return the chicken to the pot,
ladling the sauce over the chicken and serve immediately with lemon
wedges on the side. Squeeze the lemon over the chicken just before
eating.

Serve with oven-roasted thyme red potatoes or steamed short-grain brown rice + a crusty french bread to sop up the sauce.