Place the butter into a mixing bowl and beat until soft and creamy. Add the icing sugar and vanilla and continue beating until light and creamy. Gradually beat in the egg, then stir in the flour using your fingers, until to comes together. Transfer the dough to a working surface and knead for about 30 seconds. Roll the dough into a 20 cm log and flatten to 3-4 cm. Wrap in plastic and refrigerate for 30 minutes.
Preheat the oven to 180°C.
Take the dough from the fridge and cut in half, straight through the plastic. Leave one half in the fridge as you roll the other half between two sheets of baking paper. Roll to a 4 mm thickness. Using a 4 cm cookie cutter, cut the dough and place each cookie round on a lined baking sheet.
Bake for 10-15 minutes, or until a little colour develops around the edges. Remove from the oven, allow to sit for 5 minutes, then transfer the cookies to a rack to cool completely. Repeat this process with the remaining dough in the fridge.
In a small bowl combine the ¼ cup icing sugar with the water. Stir well to dissolve, squeezing out any lumps that may form. Add the walnuts, vanilla and rum and stir well to combine.
Divide the cookies in half. Spread ¼ teaspoons of the walnut mixture on the underside of half of the cookies. Set aside. Spread the underside of the other half of the cookies with ¼ teaspoon of jam, spreading evenly.
Sandwich one walnut half with the jam half, pressing together gently. Do this with the remaining cookies.
Break up the chocolate and melt in a double boiler as you spread the filling on the cookies. Alternatively melt the chocolate in the microwave.
Dip one half of each cookie into the melted chocolate and allow to harden on a sheet of baking paper or plastic. When set, store in a sealed container.