Preheat oven to 375 degrees F. Season the chicken with salt and pepper on both sides. Place the flour in a medium bowl and dredge the chicken in the mixture. Tap off any excess. Heat the oil in a large Dutch oven until almost smoking and sauti the chicken in 2 batches until golden brown on both sides. Remove the chicken to a baking sheet and finish cooking in the oven, approximately 8 to 10 minutes. Drain off all but 2 tablespoons of oil in the pan.
Add the shallots and garlic to the pan and cook until soft. Increase the heat to high, add the wine and reduce until almost dry. Add the chicken stock and continue cooking to a sauce consistency. Whisk in the butter, tarragon and peas and season with salt and pepper to taste. Remove the chicken from the oven and serve with the sauce.
For the Mushroom Duxelle: Heat olive oil in a large saute pan over high heat until almost smoking, add the mushrooms and cook until golden brown and dry. Add the shallots and garlic and cook until soft. Add the wine and cook until reduced. Add the thyme and season with salt and pepper to taste.
For the Potatoes: Preheat oven to 400 degrees F. Pour 2 tablespoons of the clarified butter into an iron skillet and heat over high heat. Arrange potatoes slices in the pan in concentric circles, overlapping the slices, drizzle each layer with butter, season with salt and pepper and spread some of the mushroom duxelle over the entire surface. Repeat to make 4 to 5 layers. Heat the pan on top of the stove until the butter bubbles.
Transfer the pan to the oven and bake until the potatoes are golden brown and soft, 20 to 25 minutes. Press the potatoes with a pot lid during the baking once or twice to keep them flat. Remove from the oven and pour out excess butter. Let sit 10 minutes before inverting onto a round platter.