Preheat the oven to 425°. In a small bowl, cover the currants with the rum and let stand for 5 minutes, stirring twice. Meanwhile, in a food processor, puree the cream cheese with the blackberry preserves, cinnamon and a pinch of salt. Scrape the cream cheese mixture into a medium bowl and stir in 1/2 cup of the walnuts and the currants with their soaking liquid.
Line a baking sheet with foil and oil it lightly. Wrap the tortillas in a kitchen towel and heat in a microwave oven until warm and pliable, about 30 seconds.
Place the tortillas on the prepared baking sheet and spoon one-fourth of the cream cheese filling just below the center of each one. Fold the left and right sides of the tortillas partially over the filling. Roll up the tortillas to form small burritos. Brush with the melted butter and bake for 5 minutes. Brush again with some butter and bake for 5 minutes longer, until golden. Remove from the oven and brush once more with butter. Sprinkle the blintzes with confectioners' sugar and let cool for 5 minutes. Transfer the blintzes to plates, sprinkle with chopped walnuts and serve at once with vanilla ice cream.