Prepare brownie mix according to package directions using oil, water and eggs and adding instant espresso coffee and vanilla to batter.
Bake in 13 x 9-inch baking pan according to package directions, but bake for only 25 minutes.
Meanwhile, make the filling. Beat the butter and brown sugar in medium bowl with electric mixer on medium speed until light and fluffy. Blend in egg, instant espresso coffee and vanilla. Stir in walnuts and chocolate until blended.
Remove brownies from oven. Carefully spread filling over partially baked brownies. Return to oven; bake an additional 15 to 18 minutes or until light brown.
To make the glaze, heat the chocolate chips and butter in small microwave-safe bowl on HIGH 30 seconds or until melted and smooth when stirred. Whisk in instant espresso coffee and milk until thin enough to drizzle. Drizzle over warm brownies. Let cool.