Season the chicken breasts generously with salt and pepper.
Sear the Chicken
In a large skillet, heat the olive oil and butter over medium-high heat. Add the chicken breasts and sear for about 5-7 minutes on each side, or until golden brown and cooked through. Remove from the skillet and set aside.
Sauté the Aromatics
In the same skillet, add the chopped shallots and minced garlic. Sauté for about 2 minutes until fragrant and translucent.
Make the Sauce
Lower the heat and pour in the heavy cream, stirring to combine with the shallots and garlic. Add the Dijon mustard, fresh tarragon, and lemon juice. Simmer for 3-4 minutes until the sauce thickens slightly.
Combine and Heat
Return the chicken to the skillet, spooning the sauce over the top. Simmer for an additional 2-3 minutes to heat through and meld the flavors.
Serve
Serve the chicken with the creamy tarragon sauce drizzled on top, alongside your choice of sides.