Heat oven to 350°F. In large bowl, beat brown sugar, butter, vanilla, rum extract and eggs with electric mixer on medium speed until light and fluffy.
Beat in flour, baking powder and salt on low speed until well blended. Stir in white chocolate chunks and walnuts. In ungreased 13x9-inch pan, spread batter evenly.
Bake 25 to 35 minutes or until top is golden brown and set. Cool completely, about 2 hours.
In small microwavable bowl, microwave semisweet chocolate chunks and oil uncovered on High 30 to 60 seconds, stirring every 15 seconds, until melted; stir well. Spread chocolate glaze over brownies. If desired, place glaze in small plastic food-storage bag and cut off small tip from one corner of bag; drizzle glaze in diagonal lines over brownies. Let stand until glaze is set. For brownies, cut into 6 rows by 6 rows.