Make the filling by stirring together the raspberries, sugar, and cardamom in a small mixing bowl, muddling the raspberries a bit as you go. When you're done the consistency should look more like jam. Line a 9 x 13 inch baking pan with parchment and set aside.
Transfer your proofed dough to a lightly floured work surface. Form the dough into a roughly even log about 14 inches long. Using a rolling pin, roll the log out into a rectangle that's about 11 inches by 14 inches.
Brush the surface of the dough with the melted butter then spread the raspberry filling on, leaving about half an inch of overhang around the edges. Carefully roll the dough back into a log, starting with the long edge. Warning: a serious amount of the filling will be pushed out during the rolling, we simply scooped it up off the work surface and put it on top of our buns before baking.
Cut the rolled dough into 12 pieces of roughly the same size. Transfer the rolls to the prepared baking pan, leaving a bit of room between rolls so they have space to rise. Cover the pan with a clean kitchen towel and allow to proof for a further 40 minutes.
Preheat the oven to 350°F. Once the buns are done proofing, pop them in the oven and bake for 20 to 25 minutes, or until lightly golden brown and the middle rolls sound hollow when you tap them with your finger. Allow the buns to cool for at least 20 minutes before frosting.
While the buns are cooling, make the frosting. In a small mixing bowl using a baking spatula, cream together the cream cheese and butter. Add the icing sugar and vanilla and mix until smooth, adding 1 to 2 tablespoons of water as you go to reach the desired consistency. If there are persistent clumps in your frosting, switch to a whisk to get the frosting nice and smooth.
Using a spoon, drizzle the frosting over the cooled buns, smoothing it with the back of the spoon as you go. Enjoy them right away, or keep them wrapped up tight for up to four days.