This will be difficult to hide because it's directly in front. Also getting an exact match on the paint may be hard too.
The repair should start by sanding the rough edges around the damage. Start with a 120 grit sandpaper and than change to 180 grit. Sand only the lip of the original finish. Start with light pressure until your sure the finish will not continue peeling off.
Once the edge is smooth it can be filled. There's many products for the patching. The most likely to endure for a long while is auto body filler (Bondo). Apply the filler and over fill slightly.
It should be ready in a few minutes. Wear a breathing mask for the next step. Using the 180 grit paper sand the filler in a circular motion, checking often that the repair is flush with the original finish. Use your fingers to check.
Wipe any dust off the surface. Apply a similar type paint and color. You will have to bring a chip from the dresser to the paint store so they can match the color.