Also goose fat for ultimate deliciousness.

Edit: Geese might disagree but they're mean.

If you are concerned about flash point, make sure it is clarified butter (Ghee). Normal butter still has a decent amount of milk that will easily burn at lower temperatures.

Butter is tasty, but can only stand low temperatures.

Both involve high temperatures, so they would call for similar kinds of fats or oils. Peanut oil stands up well to high temperatures and does well in both situations. Peanut oil isn't pro-inflammatory; in fact, due to its higher saturated fat content and lower unsaturated fat content it's more like coconut oil and animal fats in that regard.