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In a large nonstick skillet, heat the oil until shimmering. Season the chicken with salt and pepper, add to the skillet, and cook over moderately high heat, turning once, until browned, about 3 minutes per side. Transfer the chicken to a plate.
Add the butter, shallot, 1 ¼ teaspoons kosher salt, and mushrooms to the skillet and cook, stirring frequently, until the mushrooms begin to brown, about 12 minutes. Add the flour and cook until it smells lightly toasty, about 1 minute.
Add the wine, broth, and reserved chicken and bring to a simmer. Cover and cook over moderate heat until the chicken is cooked through, about 5 minutes. Transfer the chicken to a serving dish.
Increase the heat to moderately high and simmer the sauce, uncovered, until slightly reduced, about 5 minutes. Remove the skillet from the heat and stir in the minced tarragon and heavy cream. Spoon the sauce and mushrooms over the chicken and serve immediately.