Wash the chicken breasts, pat dry, and season with salt and pepper. Heat the oil over medium-high heat in a pan. Fry the chicken skin side down for 6 to 8 minutes, then flip and fry for another 6 to 8 minutes, until the internal temperature reaches 74 °C (165 °F). Remove the chicken and keep it warm on a plate.
Add the butter to the remaining oil in the pan. Add the shallots and cook over medium-high heat, stirring occasionally, for 3 to 4 minutes until softened and starting to brown. Dust with flour and cook, stirring constantly, for 1 to 2 minutes until golden and fragrant. Pour in the poultry stock while stirring constantly. Bring to a boil, then reduce heat to medium-low and simmer for 5 minutes.
Wash the tarragon, pluck the leaves, and finely chop. Stir the tarragon, Dijon mustard, and crème fraîche into the sauce. Season to taste with salt and pepper. If desired, stir in cream for a creamier texture.