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Pat the chicken breasts dry and season both sides with salt and freshly ground black pepper.
Heat 1 tablespoon of olive oil in a large, shallow skillet over medium-high heat. Add the chicken and sear until golden brown, about 4 to 5 minutes per side. Transfer the chicken to a plate.
Reduce the heat to medium and add 2 tablespoons unsalted butter. Once the butter has melted, add the chopped shallot and sliced cremini mushrooms. Cook, stirring occasionally, until the shallot is translucent and the mushrooms are golden, about 5 to 7 minutes.
Sprinkle 1 tablespoon of all-purpose flour over the shallot and mushrooms and cook, stirring constantly, for about 1 minute to form a roux.
If using, pour in 1/2 cup dry white wine, scraping up any browned bits from the bottom of the pan. Let it bubble for about 1 minute to reduce slightly.
Slowly whisk in 1 1/4 cups chicken stock. Bring to a gentle simmer and cook until the sauce begins to thicken.
Lower the heat to low, then stir in 2/3 cup heavy cream and 1 teaspoon Dijon mustard. Return the chicken to the pan, nestling it into the sauce. Cover and simmer gently for 10 to 15 minutes, or until the chicken is cooked through.
Stir in 1 tablespoon chopped fresh tarragon. Taste and adjust the seasoning with salt and pepper as needed. Serve with potatoes and your choice of greens.