Prepare balsamic vinegar reduction by adding 1 cup of balsamic vinegar to a small pot. Heat on high until the mixture boils, then turn down to simmer for 20 minutes. The reduction should yield 1/2 cup of vinegar. Store in an airtight container.
Prepare asparagus by cutting off about 1 inch of the bottom end of the asparagus, lay asparagus on a sheet pan drizzle with oil and sprinkle with salt. Roast on medium heat on a grill for about 10 minutes turning the asparagus over.
Cooking time with vary with the thickness of the asparagus. Remove asparagus from grill when done. If desired squeeze 1/2 of a lemon over the asparagus. I do not think the Olive Garden does this, but I think it really helps to bring out the flavor of the asparagus. Prepare Parmesan sauce by adding heavy cream and 2 tablespoons of grated Parmesan cheese in a small pot or skillet.
Simmer on low for about 10 - 12 minutes. The cream should reduce by 20% in volume. To prepare plate some of the cooked asparaguses on a plate, drizzle balsamic vinegar on top. Garnish with 1 tablespoon Parmesan cheese, and then serve with the sauce. Garnish with fresh chopped tomatoes if desired.
If you can only find large asparagus, you can peel the stalk with a vegetable peeler, this will peel away the tough part of the asparagus!