Combine 2/3 cup milk and butter in a saucepan; heat to 130°. Set aside.
Combine 2 cups flour, sugar, yeast, salt, and cinnamon in a large bowl; add milk mixture, and beat 5 minutes at medium speed of electric mixer. Add eggs, and stir well. Fold in currants and enough remaining flour to make a soft dough.
Turn dough out onto a floured surface; knead 4 to 6 minutes or until smooth and elastic. Place dough in a greased bowl, turning to grease top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk.
Punch dough down, and let rest 5 minutes. Turn out onto a floured surface, and roll to 1/2- inch thickness; cut with a 2- inch biscuit cutter. Place buns on lightly greased baking sheets. Cover and repeat rising procedure 1 hour or until doubled in bulk. Brush buns with egg white. Bake at 350° for 15 minutes or until lightly browned. Cool on wire racks. Combine powdered sugar and 1/4 cup milk; mix well. Drizzle over warm buns.