Preheat the oven to 375 degrees F. Season the chicken with 1/2 teaspoon salt, and pepper to taste. Heat a medium skillet over medium-high heat; add 1/2 tablespoon vegetable oil, then add the chicken, skin-side down, and cook until the skin is golden brown, about 6 minutes. Transfer to a rimmed baking sheet, skin-side up; reserve the skillet.
Toss the mushrooms with the remaining 1/2 tablespoon vegetable oil and season with salt and pepper; scatter around the chicken on the baking sheet. Bake until the mushrooms are tender and a thermometer inserted into the thickest part of the chicken registers 160 degrees F, about 25 minutes.
Meanwhile, melt the butter in the reserved skillet over medium heat. Add the flour and cook, stirring, 1 minute. Add the chicken broth and bring to a simmer, whisking. Add the wine, sugar and 1/4 teaspoon salt and simmer until the sauce is reduced by half, about 10 minutes. Season with salt and pepper.
Spoon some of the red wine sauce onto each plate; top with the chicken and mushrooms.