Put the bacon into an unheated large, high-sided skillet and cook over medium heat, stirring periodically, until the bacon is browned and crisp, about 8 minutes. Transfer the bacon to a small bowl with a slotted spoon; set aside.
Discard all but 2 tablespoons of the bacon fat in the pan. Increase the heat to medium-high. Add the mushrooms, 1/4 teaspoon salt and several grinds of pepper and cook until browned, 2 to 3 minutes. Stir in the garlic, flour and tomato paste and cook, stirring, until the tomato paste darkens a little, about 1 minute. Add the chicken broth, wine, onions, thyme, 1/2 teaspoon salt and more pepper. Bring to a boil, then let simmer until thickened, about 4 minutes.
Turn the heat down to medium, and whisk in the butter a little at a time. If the sauce tastes a little too acidic, add the sugar. Nestle the chicken and cooked bacon into the sauce, and simmer gently until the chicken is heated through, 6 to 7 minutes. (This could take up to 10 minutes if the chicken is cold, or as little as 3 minutes if it is warm.) Spoon the sauce over the chicken pieces periodically to coat completely. Toss the chicken in the sauce, remove the thyme stems, sprinkle with the parsley and serve.
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