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Preheat oven to 425°F with rack in middle.
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Pat chicken dry, then cut a deep 3-inch-long pocket horizontally in long curved side of each chicken breast half. Fill each pocket with one quarter of cheese. Season chicken all over with 1 tsp salt and 1/2 tsp pepper.
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Heat oil in an ovenproof 12-inch heavy skillet over medium-high heat until it shimmers. Place chicken, skin side down, in skillet and cook, undisturbed, until golden brown, about 4 minutes. Turn breasts over, then place skillet in oven and bake until just cooked through, 20 to 25 minutes. Transfer chicken to a platter.
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Return skillet to burner (handle will be very hot) and add water and vinegar. Boil, stirring and scraping up any brown bits and melted cheese, until sauce thickens slightly, about 3 minutes. Strain through a fine-mesh sieve into a heatproof measuring cup, then skim off excess fat. Season with salt and pepper. Serve chicken with sauce.
