Preheat oven to 400°. With a rolling pin, lightly roll pastry into a 1/8-in.-thick rectangle or square. Brush pastry with butter, then mix cardamom and sugar and sprinkle it over pastry, leaving a 1/2-in. border clear on edge. Sprinkle a single dense layer of currants on top of sugar. If pastry is sticky or stretchy, chill 15 minutes before proceeding.
Roll pastry into a tight log. Slice into 1/2-in.-thick slices and arrange, cut side up, on a parchment-lined baking sheet, pushing them back into rounds if they get irregular. Blend egg with 1 tbsp. water and brush cookies with egg wash.
Bake cookies, rotating sheets halfway through, until golden brown, 10 to 12 minutes. Mix honey with 1 tbsp. water and dab onto warm cookies to make them shiny. Let cool briefly on sheet, then lift to a rack to cool completely.