Stir 1 cup of the flour and the yeast together in a large bowl; set aside.
Heat milk, sugar, margarine or butter, salt, and nutmeg in a small saucepan until warm (120 degrees F. to 130 degrees F.) and margarine is almost melted. Add to flour mixture. Add egg. Beat with an electric mixer on low speed for 30 seconds, scraping bowl constantly. Beat on high speed for 3 minutes. Stir in chopped walnuts and as much remaining flour as you can with a wooden spoon.
Turn dough out onto a lightly floured surface. Knead in enough of the remaining flour to make a moderately stiff dough that is smooth and elastic (6 to 8 minutes total). Shape into a ball. Place in a lightly greased bowl; turn once. Cover; let rise in a warm place until double in size (about 1 hour).
Punch dough down. Turn out onto a lightly floured surface. Divide in half. Cover; let rest 10 minutes.
Divide each dough half into 12 pieces. On a lightly floured surface, roll each piece with your hands into a 12-inch-long rope. Tie each one into a loose knot, leaving two long ends. Tuck top end under roll. Bring bottom end up and tuck into center of roll. Place 2 to 3 inches apart on greased baking sheets. Mix egg white with water, if desired. Brush water or egg white and water mixture onto tops of rolls; press a walnut half into center of each roll. Cover and let rise in a warm place until almost double in size (about 30 minutes).
Bake in a 375 degree F. oven for 12 to 15 minutes or until bread is golden. Remove from pan; cool on a wire rack. Makes 24 rolls.
Prepare rolls, then store, wrapped in plastic wrap or foil, at room temperature for up to 3 days, or freeze for up to 2 months.