1. Trim any excess fat from the chicken. Rinse and pat dry with paper towels.
2. In a large sauté pan or fry pan over medium-high heat, melt the butter with olive oil.
3. When hot, add the chicken, season with salt and pepper and sauté gently, turning once, until chicken is golden and opaque throughout and the juices run clear when the meat is pierced with a knife, about 3 minutes per side.
4. Transfer the chicken to a warmed plate and keep warm.
5. Pour off any excess fat from the pan and place the pan over medium-low heat.
6. Add the shallots and sauté, stirring, until translucent, 1 to 2 minutes.
7. Add the tarragon and cream, increase the heat to medium and stir with a wooden spoon, scraping up any browned bits from the pan bottom and blending well.
8. Cook, stirring, until bubbling and thickened slightly, 2 to 3 minutes.
9. Season with salt and pepper and a squeeze of lemon juice.
10. Return the chicken to the pan and turn each breast several times in the sauce until well coated and hot.
11.Transfer the chicken to a warmed serving plate or individual plates and spoon the remaining sauce over the chicken.
12. Sprinkle with parsley and serve immediately.