For topping, in a large skillet cook the sliced onion in hot oil about 4 minutes or just until tender. Add the walnuts and the sugar. Continue to cook and stir for 5 to 8 minutes more or until onion is slightly caramelized and walnuts are lightly toasted. Stir in mustard.
Spoon some of the onion mixture onto each of the bread slices. Sprinkle with cheese. (If desired, cover bread slices and let stand at room temperature for up to 1 hour.)
Place bread slices, onion sides up, on the grill rack directly over medium-high heat; grill for 1-1/2 to 2 minutes or just until bottoms are toasted and slices are heated through. Watch bread slices carefully to avoid overbrowning. Makes 8 servings.