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Pat the chicken breasts dry and season both sides with salt and freshly ground black pepper.
Heat the olive oil in a large shallow frying pan over medium–high heat, then sear the chicken breasts for 4–5 minutes per side until golden brown; transfer to a plate.
Reduce heat to medium and add the butter; once melted, add the shallot and mushrooms and cook for 5–7 minutes until shallots are translucent and mushrooms golden.
Sprinkle the flour over the shallots and mushrooms and cook, stirring constantly, for 1 minute to form a roux.
Pour in the white wine (if using), scrape up any browned bits, and let it bubble for 1 minute to reduce.
Gradually whisk in the chicken stock, bring to a gentle simmer, and stir until the sauce begins to thicken.
Lower the heat, stir in the double cream and Dijon mustard, then nestle the chicken back into the sauce; cover and simmer gently for 10–15 minutes until cooked through.
Stir in the fresh tarragon, adjust seasoning with salt and pepper if needed, and serve with potatoes and greens.