Preheat the oven to 170 degrees or to the lowest possible temperature.

Place the flour, salt, pepper and 1 tablespoon of the thyme in a sturdy plastic food storage bag, seal and shake to combine. Beat the egg in a medium bowl. One at a time, dip the chicken pieces into the beaten egg, letting the excess drip back into the bowl, then transfer to the bag. Seal and shake to coat the chicken in the flour mixture. Transfer the coated chicken pieces to a plate.

Heat 2 tablespoons of the oil in a large, deep skillet over medium-high heat. Once the oil is hot, place the chicken pieces in the pan, smooth side down, working in batches if necessary. Cook for 5 minutes, until golden brown, then turn the pieces over and cook for 5 minutes, until golden brown on the second side. (The chicken will not be cooked through.) Transfer to an ovenproof platter and place in the oven to keep warm. (If the oven can't be set as low as 170, place the platter in the oven, turn the oven off and keep the oven door closed.)

Reduce the heat to medium and add the remaining 1 tablespoon of oil to the skillet. Stir in the onions, garlic and remaining 1 tablespoon of thyme. Cook, stirring occasionally, for 7 to 10 minutes or until the onions are translucent. Increase the heat to medium-high and cook, stirring often, for 10 minutes or until a light golden brown.

Add the wine and vinegar; cook, stirring to scrape up any browned bits, for about 3 minutes or until the liquid has reduced by half. Stir in the tomato paste, then the broth and sugar. Bring to a boil and cook for 2 minutes or until the sauce is slightly reduced. Return the chicken to the skillet, along with any accumulated juices. Turn the chicken pieces to coat them with the liquid, then cover, reduce the heat to medium and cook for 5 minutes or until the temperature of the thickest part of a chicken piece registers 165 degrees on an instant-read thermometer.

Transfer to a serving platter or individual plates, spoon the sauce over the chicken, and serve.