Position racks in the upper and lower thirds of the oven and heat the oven to 350°F.
In a small bowl, mix the walnuts, currants, brown sugar, granulated sugar, and cinnamon.
Remove the dough from the refrigerator, stand it up on its rounded edge, and cut it in half (as if halving a bagel) into two equal rounds. Return one half to the refrigerator and, if necessary, let the other half sit at room temperature until pliable enough to roll. Roll on a lightly floured surface into a 12-inch round a scant 1/8 inch thick; check frequently to be sure it's not sticking and reflour only if necessary. Sprinkle half of the filling over the dough. Roll a rolling pin over the filling to press it gently into the dough.
Cut the dough like a pie into 12 equal wedges. Roll the outside edge of a wedge up around the filling toward the narrow point. Set the roll on an ungreased or foil-lined cookie sheet with the point underneath to keep it from unrolling. Repeat with the remaining wedges, arranging the cookies 1-1/2 inches apart. Roll, fill, cut, and shape the second round of dough with the remaining filling. Sprinkle the rugelach with sugar. Bake until the cookies are light golden brown at the edges, 20 to 25 minutes, rotating the sheets from top to bottom and front to back halfway through baking. Let the cookies cool for a few minutes on the sheets and then transfer them to a rack and let cool completely.
Roll the rugelach gently and not too tightly.