1-1/2poundsrhubarbtrimmed from about 2 pounds; cut into 2" pieces
Start by making the pastry. Cut the butter into 1/2" cubes and stash in the freezer for at least 30 minutes to chill.
Once the butter is chilled put the flour, salt, baking powder, and sugar in a food processor and pulse a few times to incorporate and aerate the flour a bit. Drop in the butter and shortening, and pulse about 10 times, until the mix resembles wet sand and most of the butter is cut down to the size of peas (though don’t worry if some pieces are larger). Take off the lid of the processor, pour the cream evenly around the flour mixture, return the lid, and pulse a few more times to combine. It should hold together when pinched. Turn it out onto a lightly floured surface, form it into a ball, roll into a disc, and cover tightly with plastic wrap before stashing in the fridge for at least 1 hour, but up to 3 days.
Preheat oven to 425°.
Roll the dough out to about 1/8" thick on a lightly floured surface. Transfer to a 9-1/2" fluted tart pan. Trim off any excess dough. Crumple up a sheet of parchment paper, open it back up, set it inside the tart shell, and fill it with dried rice, beans, or pie weights. Bake for 20 minutes. Dock the shell by pricking it all over with a fork and return to the oven for 5 - 10 minutes, or until lightly golden brown. Remove from the oven and allow it to cool slightly while you get on with everything else.
While the shell is baking poach the rhubarb to soften it slightly. Combine the sugar and water in a wide, shallow saucepan and heat over medium heat just until the sugar dissolves. Tip in the rhubarb and keep the pan simmering for just about 10 - 15 minutes. You don't want them soft and fully cooked—the point is to get them started. Once they're done, carefully remove them to a plate. Keep the syrup simmering for a little while more to reduce it.
Once the tart shell is baked, reduce the oven temp to 350°.
To make the frangipane, beat the butter, sugar, and cardamom (or ginger) until light and fluffy—either in a large bowl with a wooden spoon (about 5 minutes) or in the bowl of a mixer fitted with the paddle attachment (3 minutes or so). Beat in the egg and vanilla until combined and then fold in the almond meal. Spoon into the tart shell and level off the top. Decorate the top with the rhubarb in a herringbone—or however you prefer.
Bake for 45 - 50 minutes, or until the frangipane is mostly set (a wobble in the middle is fine).
Remove from the oven, brush the top with some of the rhubarb syrup, and either rest for 10 minutes before serving or at room temperature or even chilled.
You certainly won't need all of the rhubarb syrup to brush the tart—just a few spoonfuls at most—so any leftovers can be frozen for other later uses.