In a small pan, toast the saffron over low heat, stirring, until brittle, about 2 minutes. Re move and pulverize in a mortar with a pestle or on a cutting board with the side of a large knife blade.
Put the yogurt in a large bowl.
Put the saffron back into the pan and add the cardamom and water. Bring just to a simmer, stirring.
Whisk the saffron mixture into the yogurt.
Put the yogurt mixture in a strainer lined with cheesecloth, a coffee filter, or a paper towel and set it over a bowl.
Let drain in the refrigerator for 1 hour.
Transfer the yogurt to a bowl.
Add the 1/3 cup sugar, or more to taste. With an electric mixer, beat the yogurt until slightly thickened, 2 to 3 minutes.
Pour into bowls and refrigerate until well chilled, or put in the freezer until very cold but not frozen, about 30 minutes.
Meanwhile, in a small frying pan, toast the coconut over low heat, stirring, until golden, about 2 minutes.
Serve the shrikhand topped with the mango and coconut.
