For the dough: Sift flour, salt, and sugar together three times in a large bowl; set aside.
Heat milk to lukewarm, remove from heat, and add margarine; stir until margarine is melted. Add yeast and egg yolks; mix until thoroughly combined.
Add milk mixture to the flour mixture and stir until the dough comes together. Knead with lightly oiled hands, adding more flour as necessary, until the dough is smooth and elastic. Transfer the dough to a large bowl, cover with plastic wrap and refrigerate overnight.
For the filling: Combine the walnuts and sugar in a food processor and whirl until nuts are finely chopped. Remove 1/4 cup of the walnut-sugar mixture and set aside. Add the milk and vanilla extract to the remaining nuts and process until the mixture is thoroughly combined.
Preheat oven to 350˚F. Line 2 rimmed baking sheets with parchment.
Divide dough into four equal parts and roll each piece on a lightly floured surface into a rectangle, approximately 9-inches-by-14-inches.
Spread walnut mixture evenly to 1/4 inch from the edge of the dough and roll up. Place seam-side down. Brush the rolls with the reserved egg whites. Bake for 30 minutes, until light brown on top.
For the icing: In a medium bowl combine confectioners’ sugar, margarine, and vanilla extract; add the milk slowly and stir until the icing is smooth and reaches the desired consistency.
Finely chop reserved walnut-sugar mixture. Spread a thin layer of icing on top of rolls and sprinkle evenly with walnut-sugar mixture. When completely cool, slice on the diagonal to serve.
Tip: Nut rolls make excellent gifts and freeze well if you plan to make several batches ahead of time. Simply wrap cooled rolls tightly in plastic wrap and then in a layer of aluminum foil.