Season the chicken inside and out with salt and pepper. Stuff the chicken with the tarragon.
Melt the butter with the oil in a pressure cooker over medium heat.
Brown the bird on all sides for 10 minutes. Remove the bird and drain the fat from the cooker. Add the peeled and chopped half onion and let it cook slowly for 1 or 2 minutes without browning.
Pour in the white wine and scrape the bottom of the pan with a wooden spatula to remove the cooking juices (this is called "deglazing").
Let boil and reduce by half. Pour in the vegetable broth, then put the chicken back into the pressure cooker. Close the pressure cooker and put it under pressure.
When the valve starts to whistle, count 14 minutes of cooking time. Remove from the heat and let the pressure drop slowly.
Do not open the valve suddenly, as this will damage the fibers of the meat. Set aside.