Put a baking rack in the lower third of the oven. Put a pizza stone on the rack. Preheat the oven to 350 degrees F.
For the base: Whisk the flour with the baking powder, cardamom and salt in a medium bowl. Beat the butter and sugars in the bowl of an electric mixer fitted with the paddle attachment on medium-high speed until light and fluffy, about 5 minutes. With the mixer running, slowly drizzle in the egg and beat until combined. It's ok if it looks a bit curdled. It will come back together with the addition of the flour. With the mixer on low, beat in the flour mixture until just combined.
Pat out two-thirds (about 8 ounces) of the dough into a 6-by-3-inch rectangle on a lightly floured surface. Wrap in plastic wrap and chill until firm, about 1 hour. Wrap the remaining chunk of dough in plastic wrap and refrigerate.
For the filling: Whisk 3/4 cup water with the honey and the vanilla bean seeds in a small saucepan. Add the vanilla bean pod, apricots and raisins and bring to a simmer over medium-high heat. Cover the saucepan, reduce the heat to medium and simmer until the water is mostly absorbed, about 10 minutes. Cool the mixture, remove the vanilla bean pod and then pulse in a food processor to make a textured paste.
Roll the chilled rectangle of dough on a lightly floured work surface into a 15-by-5-inch rectangle. Gently lift it into a 13-by-4-inch rectangular fluted tart pan with a removable bottom. If the dough cracks, just press it back together with the flats of your fingertips. At the short ends the dough will come up to the top of the pan, and on the long sides the dough will come about halfway up.
Spread the apricot filling over the dough in an even layer. Crumble the reserved chilled dough over the top, squeezing it flat between your fingers as you do. It is ok if some of the filling shows through, but make sure the dough is actually touching the sides of the pan. Scatter the pistachios over top, pressing in lightly where they land on the uncovered filling. Bake on the preheated stone until the top is deep golden brown, 30 to 35 minutes. Cool completely on a rack, about 2 hours, before unmolding and slicing into 12 bars.
Cook's Note: To cook in an 8-inch round pan: Butter the bottom of an 8-inch round cake pan and line with parchment paper. On a lightly floured work surface, press two-thirds of the dough into a 5-or-6-inch round, and then transfer it to the pan and press it to the edges and up the sides. Wrap the remaining chunk of dough in plastic wrap and chill dough until firm, about 1 hour. Proceed as above with the filling and the topping.