Preheat your oven to 400 degrees. Add the oil to the skillet and place over medium-high heat.
While the oil is heating, put the chicken thighs and drumsticks in a bowl and season with the salt, pepper, and thyme.
When the oil is hot, add the chicken to the skillet skin-side down, alternating between thighs and drumsticks.
Let the chicken cook for about 10 minutes, or until the skin is crispy and golden-brown.
Flip all the chicken pieces over and remove the skillet from heat.
At this point, add the potatoes and onions. Make sure to fill in the spaces under and between the chicken pieces.
Once they’re all squeezed in there, pour the chicken broth and buttermilk over the top and put the skillet into the oven for about 40 minutes.
Top with the tarragon leaves and serve.