Jasmine extract is a Thai staple that adds a floral note to this creamy rice pudding.
Heat milk in a medium saucepan until hot but not boiling. Add tea leaves or essence. Remove from heat, cover, and let steep 20 minutes. Strain milk if using tea leaves; set aside.
Bring 3 cups water to a boil in a medium saucepan. Stir in rice and salt, reduce heat to low, cover, and cook until rice is tender and liquid has evaporated, 15 to 20 minutes. Stir milk, butter, and 1/2 cup sugar into rice. Cover and cook over medium-low heat, stirring occasionally, until sugar is dissolved and the pudding is thick, about 30 minutes.
Spoon pudding into dessert dishes, and cover with plastic wrap to prevent skin from forming; set aside in refrigerator. Place remaining cup sugar in a medium skillet over medium-low heat. Cook, without stirring, until dark amber in color, about 15 minutes. Pour caramel over top of each pudding. Serve immediately or at room temperature.