Combine water, sugar and yeast. Stir to dissolve yeast and let stand until bubbly, about 5 minutes. Combine 1/3 cup milk, 2 TBS butter and salt in a small saucepan. Heat over low heat until just warm (105° -115°). Butter may not melt completely. Remove from heat.
Fit the food processor with a steel blade. Measure the flour into the work bowl. Add the yeast mixture. Process on/off until mixed, about 5 seconds. Turn on the processor and add milk through the feed tube. Process until mixed, about 30 seconds. If necessary, add more of the remaining milk to make a very soft dough. Cover the processor and let dough stand in work bowl at room temperature for 15 minutes. In a small bowl, mix walnuts, remaining 2 TBS butter, brown sugar and cinnamon.
Turn the dough onto a lightly floured surface. Roll out into a 10 inch square, and spread the walnut mixture over the dough. Roll up dough jelly roll fashion. Pinch the seam to seal and arrange in a circle on a large greased cookie sheet. Pinch the ends of the dough together to seal. With scissors, cut 2/3 of the way through the ring at 3/4 inch intervals, leaving the center of the ring intact. Gently lay each section on its side, cut side up to show filling. Brush with oil, cover with plastic wrap and refrigerate overnight.
Heat oven to 375°. Let ring stand at room temperature for 15 minutes prior to baking. Bake until golden, 20-25 minutes. Remove from cookie sheet and cool. Sprinkle with powdered sugar. Makes one coffee cake.