Preheat your oven to 450°F.
Ensure your chicken is not ice cold. Drain any excess juices and pat the chicken completely dry with paper towels.
With clean hands, carefully separate the skin from the breast meat and thighs without tearing the skin.
In a small bowl, combine the butter and olive oil. Microwave at 50% power until melted.
Add the chopped tarragon, salt, and pepper to the melted butter mixture and stir to combine.
Using a spoon, carefully distribute the butter mixture under the skin of the chicken, covering the breast meat and thighs. Also rub some of the mixture over the outside of the chicken.
Create a bed of sliced onions in a roasting dish.
Place the garlic halves and about 3 sprigs of thyme inside the chicken cavity.
Squeeze the lemon halves over the chicken, then place the squeezed lemon halves inside the cavity.
Fold the wings under the body of the chicken and tie the legs together with kitchen twine.
Place additional thyme sprigs around the chicken.
If adding vegetables, toss them with a drizzle of olive oil, salt, and pepper and arrange around the chicken.
Place the chicken in the preheated oven and immediately reduce the temperature to 425°F.
Bake for approximately 75-90 minutes, basting every 30 minutes with the pan juices.
The chicken is done when it's golden brown and a thermometer inserted into the thickest part reads 165°F.
Once cooked, allow the chicken to rest for 20 minutes before carving to ensure the juices are reabsorbed.