Preheat the oven to 200°C (180°C fan).
In a small pan, heat the olive oil, add the diced shallot and cook for 2–3 minutes. Stir in the minced garlic for another 30 seconds. Remove from heat and let cool. Mix into the softened butter with the chopped tarragon, lemon zest, mustard, salt, and pepper.
Pat the chicken dry and loosen the skin over the breasts and thighs. Rub the herb butter under the skin and over the chicken. Stuff the cavity with the halved lemon and the leftover sprigs of tarragon. Season generously with salt and pepper.
Place the chicken in a roasting tin and roast for 60–75 minutes or until juices run clear, basting halfway through.
Once the chicken is cooked, transfer it to a plate to rest. Immediately pour the cream into the hot roasting tin, stirring well to deglaze and loosen the caramelised bits at the bottom. Return the tin to the oven for 4-5 minutes, just until the sauce is heated through and slightly thickened. Stir well to combine before pouring it into a small serving jug.
Taste the sauce and adjust seasoning with salt and pepper if needed. Carve the chicken and serve with the creamy sauce poured over the top.
