Preheat oven to 325 degrees F (165 degrees C). Line a baking sheet with aluminum foil.

Place potato wedges in a bowl; sprinkle thyme, cayenne, salt, and black pepper over the top. Drizzle melted duck fat over potatoes and toss to coat.

Transfer potatoes, skin-side down, to the prepared baking sheet. Season wedges again with salt.

Bake in the preheated oven for 40 minutes. Increase oven temperature to 450 degrees F (230 degrees C). Turn potatoes onto their sides and bake for 10 minutes. Flip over and cook on the other side until golden brown, about 10 minutes more.