In a large skillet cook onions in hot oil over medium-high heat for 5 to 7 minutes or until onions are tender and just starting to brown, stirring frequently. Remove from heat; cool slightly.
To assemble, lightly spread one side of 4 bread slices with honey mustard. Top with the onion slices, tomato slices, shredded lettuce, and basil. Spread one side of the remaining 4 bread slices with Brie cheese. Place the bread slices, cheese sides down, on top of sandwiches.