Pat dry the chicken thighs and drumsticks with a paper towel. This will help give the chicken a good sear as well as help the salt and pepper stick to it. Season chicken with salt and pepper on both sides.
Melt butter in a large skillet over medium-low heat. Brown chicken on both sides, leaving it to cook for 5 minutes on each side. Remove the chicken pieces from the skillet and transfer them onto a plate to rest.
Using the same skillet, add the flour to the remaining oils and whisk for a minute. It should start to create a roux which is a light brown paste with a bit of a nutty aroma.
Return the chicken you set aside back into the skillet and allow it to come back to a gentle boil. Cover with a lid and simmer on low for 25-30 minutes until the chicken is cooked through.
At the end, stir in the heavy whipping cream and allow it to thicken up a bit. Do a little taste test and season with salt and pepper. Garnish with fresh thyme or parsley and enjoy!