Preheat the oven to thermostat 6-7 (200°C). Butter a round or square tart tin with 15 g of the butter and line it with the puff pastry.
In a bowl, cream 125 g softened butter with the sugar until light and fluffy. Beat in the egg yolks one by one, then fold in the ground almonds, flour, and Armagnac to form a smooth frangipane. Spread evenly over the pastry base and bake for 30 minutes until golden.
Meanwhile, rinse and pat dry the figs, then slice lengthwise into quarters or sixths depending on size. Sauté briefly in the remaining 40 g butter over medium heat. Toast the pine nuts in a dry pan for a few minutes until golden.
Let the baked tart cool slightly. Arrange the fig slices upright, pointy side up, on top. Warm the currant jelly until melted and lukewarm, then brush over the figs. Scatter with toasted pine nuts.
Serve warm or chilled for a sophisticated dessert that highlights seasonal figs.
