Put the rice in a bowl and gently rinse with three or four changes of water, until the water runs fairly clear. Fill the bowl halfway with cold water and let the rice soak at room temperature for 30 min. to soften the kernels. Drain.
Heat a 3-quart saucepan over medium-high heat; pour in the oil and swirl it around. It should appear to shimmer; if not, continue heating until it does. Add the cumin, cardamom, cinnamon, and bay leaves, which will sputter and crackle and smell aromatic. After 20 to 30 seconds, add the onion and cook, stirring frequently, until the onion turns golden brown, about 5 min.
Add the drained rice and toss gently with the onion and spice mixture. Pour in 1-1/2 cups cold water and sprinkle on the salt and saffron. Stir the rice just once to incorporate the salt and saffron. Leaving the pot uncovered, bring the water to a boil over the same medium-high heat. After about 3 min., when much of the water has evaporated or been absorbed (if you move some of the rice with a fork, the water should look like it comes about halfway up the rice), stir the rice once more to bring the partially cooked layer from the bottom of the pan to the surface. Cover the pan with a tight-fitting lid and turn down the heat to the lowest possible setting. Cook covered for another 10 min. Without removing the cover, turn off the heat and let the pan stand off the heat, undisturbed, for another 10 min.
Remove the lid and add the mint. Fluff the rice with a fork to let the steam escape and to incorporate the mint leaves. Remove the cinnamon sticks and bay leaves and instruct your guests not to eat the cardamom pods.