Potato Tahdig

1/4 tsp. saffron threads, finely ground
3 cups basmati rice
Kosher salt
1/3 cup plus 1/4 cup vegetable oil
2 russet potatoes (about 1 1/4 lb. total) peeled, sliced into 1/2"-thick rounds

Stir saffron and 2 Tbsp. lukewarm water in a small bowl; set aside to bloom.
Meanwhile, place rice in a strainer or fine-mesh sieve and rinse with lukewarm water, swishing around with your hands to get rid of excess starch, until water runs clear. Cook rice in a large pot of boiling generously salted water for 7 minutes. Check one grain; it should be soft on the edges but firm inside. You should be able to break it into two using your fingertips, but the grain should still be firm in the middle. If not, cook 1 minute longer and check again. Drain rice and rinse with cold water to stop the cooking.
Wipe out pot. Pour 1/3 cup oil into pot and heat over medium. Add saffron mixture, tilting pot to make sure it covers the bottom.
Arrange potato slices in an even layer in bottom of pot. Season with 2 tsp. salt. Return rice to pot on top of potatoes in a mound (so it’s higher in the middle than the edges) without pressing or packing it down. Using the handle of a wooden spoon, make 5 holes in rice, going all the way down to the potato layer, so steam can escape. Pour 1/2 cup water around outer edge of potatoes. Cover lid with a clean dish towel, tying the ends up over the handle so they don’t catch fire, then cover pot with lid. Cook over medium-high heat until steaming, 7–10 minutes (check after 7 minutes; if you don’t see steam, cover and continue to cook).
Pour remaining 1/4 cup oil evenly over rice, cover pot, and let steam over low heat until rice and potatoes are cooked through, 30–40 minutes. Taste to make sure rice is fully cooked; if not, add 1/4 cup more water, then let steam another 10 minutes.
Spoon rice onto a platter. Using a spatula, lift crispy potatoes and rice off bottom of pot (this is the tahdig) and arrange, browned side up, over rice.