Sprinkle the chicken with salt and pepper (to taste).
Heat the oil in a dutch oven or a heavy large skillet over medium heat.
Add the chicken and cook just until browned, 4 to 8 minutes.
Transfer the chicken to a plate and set aside.
Melt the butter in the same dutch oven.
Add the onion and saute over medium-high heat until tender, about 2 minutes.
Add the mushrooms and garlic, and saute until the mushrooms are tender and the juices evaporate, 12 - 15 minutes.
Add the wine and simmer until it is reduced by half, about 4 minutes.
Stir in the mascarpone and mustard, stirring until the mascarpone all melts.
Return the chicken and any accumulated juices to the skillet.
Stir in the tarragon.
Simmer, uncovered, over medium-low heat until the chicken is cooked through and the sauce thickens, about 2 - 8 minutes.
Pairs well with fettucine.