Heat a few tablespoons of oil in a Dutch oven over high heat. Sprinkle the chicken pieces on both sides with salt and pepper. Place the chicken in skin-side down and cook until golden brown. Turn the chicken over and brown the other side. Remove the chicken to a baking sheet or plate and set aside.
Add the mushrooms, season with salt and pepper and saute until golden brown. Remove the mushrooms from the pan. Add a touch more canola oil and add the shallots and cook for 30 seconds. Add 1 tablespoon of the tomato paste and cook for 1 to 2 minutes. Add the white wine and cognac and reduce for 4 to 5 minutes. Add the stock and bring to a boil. Add the mushrooms and nestle the chicken into the sauce, skin-side up. Reduce the heat to a simmer, cover and cook until the chicken is cooked through, about 15 minutes.
Once the chicken is cooked, remove it from the pan and put on a plate. Taste the sauce. Add 1 more tablespoon tomato paste and whisk to combine. Add the butter to enrich the sauce. Check the seasoning. Add the tarragon and stir to combine.
Serve the sauce with the chicken and some of the Creamy Polenta with Gruyere and Parmesan. Garnish with a couple sprigs of chervil.
Add the stock to a medium saucepan and bring to a boil over high heat. Add the polenta in a fine stream, whisking constantly with a wire whisk. Once all the polenta has been added, reduce the heat to low and cover the pan. Stir the mixture, using a wooden spoon, every 5 minutes and continue cooking until all the liquid has been absorbed and the mixture is very creamy, 10 to 15 minutes.
Remove from the heat and stir in the heavy cream and cheeses until combined. Taste and season with salt and pepper.