How do I add thickness to my cardboard cut-out image-as-plane? Greetings fellow Blenderers. This is a continuation of the issue first presented here.
I have a very nice 2d plane that has both my PBR cardboard texture as well as the single image with alpha-channel. Now, I'd like to solidify it. Currently, this is the result:

Now, I completely understand why there is no center. It's just a flat plane and there is no geometry to extrude, per se. My question then becomes: How do I do that? Or, if the answer is (it's too complicated), then how do I fake it?
Thanks in advance for your attention. Hang in there. Like that sun. All smiling and happy. What's his con?..
UPDATE
Okay, I have an idea of a workflow that will work, but I have to get over one really big sticking point:

Keep the existing material because that part works
Go into Illustrator and turn the overall shape of the sun into a single SVG
Import the SVG
Object - Convert to Mesh from Curve
Solidify
Marvel at the truly ghastly default geometry that has been created

Become confused as I can't effectively remesh the resulting geometry so that it has nice, square shaped UV maps.

So yeah, List Item #7 is where I am currently stuck on.
Be great.
Here is a link to my source file. I have pain-stakingly broken out my process flow into individual layer/collections. It should be quite self explanatory once you open it. It's basically a time-lapse of my activities with the shape, trying to make the shape do what I need to eventually receive the texture. I am now amazed at how hard this is.
Update on AM of 7/28 / Thank you for the overnight responses. Specific replies:

Susu: I actually did that, but the resulting mesh (using either fill method) was similarly bad looking, though it was still be best result given all the other bad ones I encountered.
Moonboots: That's actually what I want to avoid. I have a number of 2d images I'd like to convert to 'cutouts' and manually drawing points takes far too much time and effort.
Jachym: Sadly, that doesn't work. When it comes time to apply my texture, I encounter all kinds of display problems. Something about the botched geometry is deforming the results.
General Reflection on this: The 2D import behavior is just wretched. I come from a 2D background where SVG's have mostly behave themselves and worked fine in whatever workflow I happen to find them in. But Blender needs to an intervention.
Glorious (7/29 Afternoon): I have definitely decimated. Included above is a link to the source file. Thanks for looking! Also, projection mapping didn't work. There's something wonky in the geometry that caused it to look mangled.

Thank you all very much for your contributions.
Current Key Observation: I'm having the darnedest time removing the side geometry from my little sun shape. I can't figure out a proper way to auto-select it without completely butchering the overall geometry. What kills me is that I know I did it last night, but I can't replicate it.
I realized that - once again - I'm overthinking this. With more time away, the solution came to me.
In my above question, I identify Step #7 as the issue, but it doesn't have to be. Instead, I realize I should ignore all that goofy geometry. All that matters is I have my ugly extrude with terrible faces.
What I have is that extruded shape set at 0.009 m Z-depth (thin like cardboard). Then I put my 2D plane on it. As long as the material is close to the color of the cardboard (or dominant 2d image color), it will look just fine.
My Current Understanding of Workflow:

Process image to ensure that it has necessary alpha channel. Export a PNG.
Pull image into Illustrator and use the Image Trace option to create a simple outline around the imagery. Export a SVG.
Import Image as Plane; apply additional textures as necessary (if overlaying texture on top of corrugated cardboard texture as I am).
Import SVG as Curves; extrude to needed depth; turn to a mesh.
Place the flat 2D image 'on top of' the 3D mesh that will become the cardboard's faked-depth.
Enter orthographic mode to look at the model head-on. Make final adjustments to ensure the SVG and PNG 'match' locations, as my tests have shown that it won't perfectly match at first. Simply nudge it in ortho and you get your results.


Final note: For whatever reason, when I do this, the 2D image clips into one side of the mesh geometry. Simply adjust the placement (or flip the model over, as I have). This seems to fix it.
Thanks to all of you on my little mini-rollercoaster ride. I will treasure this moment of feeling like I figured something out... because Blender amirite..