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In a medium bowl, mix together the rice flour, 1 teaspoon salt, and 1/4 teaspoon pepper. Add the chicken and toss to combine.
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In a large heavy pot with a lid or Dutch oven over medium-high heat, warm the oil until hot but not smoking. Working in batches to avoid crowding, carefully place the chicken in the oil and brown on both sides, 2 to 3 minutes per side, adding more oil if needed. Transfer the chicken to a plate.
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Add the onion and mushrooms to the pan, season with ½ teaspoon salt and cook, stirring often, until the onions are soft and begin to brown and the mushrooms have released all their liquid, about 8 minutes.
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Deglaze the pan with Cognac, then add the wine, broth, tomatoes, and herbes de Provence. Stir to combine, bring to a boil over high heat, then reduce the heat, return the chicken to the pan and simmer, turning once or twice, until the mixture thickens to a slightly watery gravy, 3 to 5 minutes. Stir in the butter and half of the parsley, season to taste with salt and pepper, garnish with the remaining parsley, and serve over individual bowls of polenta.