Preheat the oven to 200C/fan 180C/ gas mark 6 and grease and fully line a 22cm square cake tin.
Whisk the cream cheese and sugar together until smooth then beat in the vanilla extract, coffee and ground cardamom. You will have a soft floppy mixture.
Unroll the croissant dough and lay out across a lightly floured work surface. Spread the coffee cheese mixture over the croissant dough to cover all but 2cm down one long edge.
Taking the other long edge, roll the dough up carefully into a pinwheel - the edge with no cheese will be the seal underneath.
Using a sharp knife, cut the dough into 12 equal rounds. Place them in rows into the square cake tin until it is full but not too tightly packed.
Bake in the oven for 18-20 minutes until golden brown and risen.
Remove and cool for 10 minutes in the tin before lifting out, using the parchment paper and laying onto a wire rack.
Mix the icing sugar with the cold coffee, a little at a time, until a runny icing is formed. Drizzle it all over the buns.