Season the chicken pieces with salt and freshly ground black pepper, then dredge in the flour to coat.
Heat a large, deep frying pan over a medium heat. Add the oil, and a third of the butter and heat until the butter is foaming.
Add the chicken pieces and fry skin-side down for 1-2 minutes until golden-brown, then turn the chicken and fry on the other side for another 1-2 minutes.
Meanwhile, heat a separate frying pan over a medium heat and add another third of the butter. When the butter is foaming add the button mushrooms and fry, stirring occasionally, until golden-brown all over.
Add the mushrooms to the chicken pan.
To the pan used to fry the mushrooms, add the remaining third of the butter, the shallots and the caster sugar and fry for 2-3 minutes until the shallots are golden-brown and caramelised.
Add the wine and bring to the boil, scraping the base of the pan with a wooden spoon to dislodge any brown bits.
Pour the shallots and wine mixture over the chicken, then add the stock, tomato purée and whole tarragon sprigs. Bring to the boil, then reduce the heat to simmer for 45 minutes to one hour, or until the chicken is completely cooked through and the sauce has reduced slightly.
Add the tomatoes, flatleaf parsley and the chopped tarragon and season, to taste, with salt and freshly ground black pepper. Simmer for a further two minutes.
Meanwhile, for the mashed potatoes, place the potatoes into a pan of salted water and bring to the boil. Reduce the heat and simmer for 12-15 minutes, or until the potatoes are tender.
Drain the potatoes and return to the pan. Place over a low heat for one minute to drive off any excess moisture.
Mash the potatoes well, then add the butter and milk, beating well to form a smooth mash. Season, to taste, with salt and freshly ground black pepper.
To serve, place two pieces of chicken onto each plate. Drizzle the sauce over each and place a dollop of mashed potatoes alongside.