Need Help Writing Blurred Edge Effect Shader On A Polygon2D

SingleShot

I have a Polygon2D that is a square with rounded corners. I'd like to give it a simulated shadow effect and I'm thinking I can use a shader to apply a gaussian blur to the edges of an identical polygon behind it. However, all my attempts have failed and I'm having trouble finding any examples - most examples apply a blur to the background, but I want the edges of the actual polygon to blur. I'd appreciate any help in producing an example shader. Thanks!

Example of what I want (drawn in an art program):
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SingleShot

I've decided to just use textures for the shadow/blur effect, but would like to have a general purpose shadow/blur for arbitrary polygon shapes, so am still interested in an example.