Combine the currants and cider in a small bowl; set aside to soak. Meanwhile, whisk the flour, oats, pie spice, baking soda, 1/2 teaspoon salt and the cloves in a medium bowl.
Beat the butter and brown sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Add the molasses and egg; beat until combined, about1 more minute.
Drain the currants, reserving the cider. Reduce the mixer speed to medium low; beat in the flour mixture in 2 batches, alternating with the reserved cider. Continue beating until the dough starts coming together, about 2 more minutes. Beat in the currants and walnuts. Cover and refrigerate until firm, at least 1 hour.
Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Coat 2 baking sheets with cooking spray. Drop rounded tablespoonfuls of dough 2 inches apart onto the baking sheets. Bake, switching the pans halfway through, until the cookies darken around the edges, 18 to 20 minutes. Let cool 10 minutes, then transfer to racks to cool completely.
Whisk the confectioners' sugar, lemon juice, vanilla and a pinch of salt in a bowl, then whisk in 1 tablespoon water. Drizzle on the cookies and let set about 10 minutes.