1. Sauté first 3 ingredients in hot oil 10 to 12 minutes or just until vegetables are tender and onion slices begin to turn golden.
2. Trim and discard rind from Brie. Cut cheese into 1/2-inch cubes.
3. Layer a lightly greased 13- x 9-inch baking dish with half each of bread cubes, onion mixture, Brie cubes, and Parmesan cheese.
4. Whisk together eggs and next 4 ingredients; pour half of egg mixture evenly over cheeses. Repeat layers once. Cover and chill at least 8 hours or up to 24 hours.
5. Bake at 350° for 45 to 50 minutes or until lightly browned on top and set in center.
*2 cups (8 oz.) shredded Swiss cheese may be substituted.