Simmer the fig fillings (except lemon juice) in a saucepan, stirring occasionally until figs are soft and the sauce is syrupy (about 30 minutes).
Cool slightly and transfer to a food processor. Add lemon and purée the figs.
Whisk flour, baking powder, and salt in a medium bowl.
Beat butter and sugar until light yellow in color, then add in the eggs and vanilla; beat until combined.
Stir in the flour mixture.
Divide the dough into two parts; put the other half into the freezer.
Put the other half in a 9-inch square baking pan lined with parchment paper, pressing the dough into an even layer.
Bake the crust until golden (20 minutes).
Spread the fig filling.
Take out the cold dough from the freezer and lay it over the filling to form as a top crust.
Bake for another 25 to 30 minutes, until golden brown.
Let cool completely, then cut into squares and serve.