Half the appeal of making this festive coffee cake is rolling the balls of dough in cinnamon sugar. The balance, of course, is pulling it apart and polishing it off.
In a large mixing bowl, dissolve the yeast in the warm water. Warm 1 cup of milk and stir in 4 tablespoons of the butter. Add the milk mixture, 1/2 cup of sugar, the salt, and 1 cup of flour to the yeast. Stir well. Add the remaining flour, 1 cup at a time, stirring until the dough feels stiff. Then turn the dough into a lightly floured board and knead for 8 to 10 minutes, adding extra flour if needed. Put the dough in a greased bowl, cover, and let it rise until it has doubled in bulk (about 1 to 1-1/2 hours).
Grease a 12-cup bundt pan. Melt the remaining butter and pour it into the pan. Add the walnuts. Combine the rest of the sugar with the cinnamon and sprinkle 1/4 cup of the mixture over the nuts.
Punch down the risen dough. Then break off small pieces, form balls, and roll them in more of the cinnamon sugar (a great job for kids). Layer the balls in the pan. Cover and let rise until doubled in bulk (about 45 minutes).
Heat the oven to 350 degrees. Sprinkle any remaining cinnamon sugar on the dough balls, then bake for 30 to 35 minutes or until golden brown.
Immediately invert the coffee cake onto a plate and let it cool for a few minutes. Finally, stir together the confectioners' sugar and the 4 teaspoons of milk and drizzle the icing over the cake. Serves 10 to 12.