Season the whole chicken all over with kosher salt and black pepper.
Heat the neutral oil and 1 tbsp unsalted butter in a wide, heavy skillet or Dutch oven over medium-high heat. Add the chicken skin-side down and brown until deep golden, 6–8 minutes; flip and brown the second side, 3–4 minutes. Transfer the chicken to a plate.
Pour off excess fat, leaving about 1 tbsp in the pan. Reduce heat to medium, add the shallot, and cook until translucent, 3–4 minutes. Stir in the garlic and cook until fragrant, 30 seconds.
Pour in the dry white wine, scraping up browned bits. Simmer until reduced by about half, 2–3 minutes.
Add the low-sodium chicken stock and the fresh tarragon sprigs; return the chicken to the pan skin-side up and bring to a gentle simmer.
Cover, reduce heat to low, and cook until the chicken is just tender and cooked through, 20–25 minutes; the juices should run clear or an instant-read thermometer should read 175°F at the thigh.
Transfer the chicken to a warm plate and loosely cover. Discard the herb sprigs. Increase heat to medium; whisk in the Dijon mustard and the heavy cream and simmer until the sauce lightly thickens and coats a spoon, 3–5 minutes.
Stir in the fresh tarragon leaves and the lemon juice. Taste and adjust seasoning as needed.
Off the heat, whisk in the remaining 1 tbsp butter to gloss the sauce.
Return the chicken to the pan, spoon sauce over the pieces, and warm for 1–2 minutes. Serve immediately with the pan sauce.