In a shallow bowl, combine the flour, 1/2 tsp salt and 1/4 tsp pepper.
Heat the oil in a large skillet over medium heat. Coat the chicken in the flour mixture then cook until golden brown and cooked through, 6 to 7 minutes per side; transfer to a plate.
While the chicken is cooking, prep your mushrooms.
Melt butter in a small saucepan over medium heat. Add garlic and as soon as garlic is fragrant, after about 1 minute, add mushrooms. Be sure not to crowd the pan - the mushrooms need room to caramelize.
Sauté for about 5 minutes, until golden brown. Remove from heat and set aside.
Add the wine and shallots to the skillet and cook until the shallots are just tender, about 2 minutes. Add the cream, mustard, tarragon and 1/4 cup chicken broth, and cook, whisking occasionally, until the sauce slightly thickens, about 2 minutes.
Whisk any accumulated chicken juices from the plate into the pan, and stir in the sautéed mushrooms.
Add the chicken back to the pan. Over low heat, spoon sauce over the chicken, for 1-2 minutes to gently reheat the chicken.
Serve immediately with an extra sprinkle of fresh tarragon and fresh parsley.