NOTE: About those onions -- 10 cups is somewhere between 3 and 4 very large onions. The recipe I took this from called for sweet onions. First time I made it, I used Vidalia -- and thought they made it a little bland. Next time I tried spicier brown onions, and thought it had a lot more flavor.
In a LARGE skillet, melt butter. Add brown sugar and stir until it dissolves.
Add onion and garlic to skillet. Stir to coat with butter/sugar mixture. Cover and cook over medium heat for at least 30 minutes; you can cook them much longer and it will only make it sweeter.
Uncover and add vinegar to onion mixture. Cook, uncovered for at least 10 minutes. (I found I cooked it nearly 45 minutes, to try to reduce some of the liquid.).