To make the tamarind-lime curd, beat together the coconut oil, 3 eggs, ½ cup sugar, lime juice, tamarind paste, and a pinch of salt (about ⅛ teaspoon) in a small saucepan.
Cook the mixture over low heat, stirring constantly, until it reaches 165 F. (The curd should have thickened to a custardy consistency at this point.)
Strain the curd, discard any solids, and refrigerate it while you prepare the cookies.
Preheat the oven to 350 F.
Combine the flour, baking powder, 1 teaspoon salt, lime zest, and allspice in a large bowl.
In a medium bowl, cream the remaining ¾ cup sugar with the butter, remaining 2 eggs, and vanilla extract, then add this mixture to the dry ingredients and stir until a dough forms.
Work the dough briefly with your hands on a clean surface, then form it into a ball.
Pinch off a piece of dough that's about the size of a large marble, and roll it into a sphere. Place the piece of dough on a cookie sheet and press your thumb into the center to form an indentation. Repeat until all the dough is used.
Spoon the tamarind-lime curd into the center of each cookie.
Bake the cookies for 15-20 minutes or until they are toasted on the bottom and slightly soft on top.