Preheat the oven to 350 degrees. Thaw the puff pastry dough. If using a rectangle tart tin, on a lightly floured work surface, roll out the dough to fit the length of your rectangular tart pan.
Place the dough in your tart tin, press on the bottom and up the edges of the pastry pan. Using a rolling pin, roll it along the top edges of the tart pan and remove the little excess pastry dough. With a fork, pock the puff pastry dough all over the bottom and bake in the oven for about 24 minutes or until lightly golden brown. Remove from the oven and increase the heat to 400 degrees.
While the puff pastry tart bakes, in a large bowl, whisk the eggs until combined. Add the almond flour, brown sugar, butter, and salt and mix with a wooden spoon (or a stand mixer if preferred) until smooth and well combined.
In a small mixing bowl, combine the sugar and ground cardamom.
When the pastry is done baking, gently press the dough in the bottom of the pan. Spread a layer of frangipane over the bottom of the pastry and up the sides.
Arrange the plum slices over the top of the filling and sprinkle the cardamom sugar over the top.
Return the pan to the oven and bake the frangipane until it is set and golden brown in spots, about 30 to 35 minutes. If the edges begin brown before the frangipane is set, cover the edges loosely with foil.
Remove from the oven and transfer to a cutting board until slightly cooled. Serve the frangipane tarts warm or at room temperature (and it’s still good the next day).