Morning or Evening
Rinse the rice very well in a sieve under running water. Dump into a 3-quart (3-L) saucepan and add 1 teaspoon salt and enough water to cover the rice by 2 inches (5 cm). Bring to a boil, stirring occasionally to keep grains from sticking to the pan; lower the heat and simmer briskly for 10 minutes.
Put the remaining 1 teaspoon salt, the saffron, and 2 tablespoons of the butter (cut into pieces) in a heatproof bowl. When the rice has simmered for 10 minutes, ladle ¼ cup (60 ml) of the hot cooking water into the saffron-butter bowl. Drain the rice in the sieve and cool under running water. Set aside over the cooled saucepan, covered with the pan lid, in the refrigerator if doing this in the morning.
With a fork or whisk, stir the yogurt into the saffron butter, whisking until smooth. In the saucepan, melt the remaining 2 tablespoons butter over medium heat, swirling the pan to coat the bottom and partway up the sides. Gently spoon in the parboiled rice, letting it mound up in the center in a cone shape, then drizzle the saffron butter–yogurt mixture evenly all over the rice. Cover the saucepan, with a clean kitchen towel underneath the lid and its corners folded over the top, and continue to cook over medium heat for 15 minutes.
Wet a towel and lay it out on a heatproof surface. Move the hot pan to the towel and let stand for 5 minutes. Uncover, fluff, and spoon out the rice, dislodging the crusty bottom layer (the tahdig) and equitably distributing the pieces. If you’d like, sprinkle with the mint. Serve.