Add 2 tablespoons oil and 1 1/2 tablespoons butter to a large frying pan over medium-high heat.
Season the trimmed chicken pieces very well with salt and pepper.
Add to the pan, skin side down, and fry until deep-golden brown; turn and repeat on the non-skin side.
Heat a heavy-bottomed Dutch oven over medium-high heat, then add 1 tablespoon oil.
Fry the salt pork lardons until they have rendered their fat and are crisp.
Then add the shallots and cook until golden.
Transfer the chicken from the frying pan to the Dutch oven with the lardons and shallots, and continue cooking over medium heat.
Pour off most of the drippings from the frying pan, return it to the heat, and add the remaining 1 1/2 tablespoons butter.
Fry the mushrooms for about 3 minutes.
Add the Madeira and flambé.
Add the demiglace, cream, and tarragon, and mix well.
Cook until it begins to bubble and form a sauce.
Pour the mushroom sauce over the chicken and simmer, partially covered, for 20-25 minutes.
To serve, pile the chicken in a serving bowl, cover in sauce and garnish with truffle shavings.
If you aren't using the truffle, sprinkle with additional tarragon or parsley.