Make the caramel: In a heavy-bottomed saucepan over medium heat, add the palm sugar and white sugar. Add 5 tablespoons of the brewed Thai tea. Add the cream of tartar (it helps prevent crystallization). Stir with a rubber spatula continuously until the sugars dissolve, 1 to 2 minutes. Continue stirring until the caramel sauce begins to foam vigorously, about 3 to 3½ minutes. Then add the shortening, coconut milk, and salt, stirring to combine. Stir for 1 to 2 minutes, then lower the heat and stir occasionally, until it reduces to about the consistency of condensed milk, about 5 minutes. Remove from the heat and let cool. It will thicken as it cools.