Can posts show branded link instead of just photo with short link?

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My posts come out and there’s the image and the link is so small. No one is engaging with these posts but when I post the link directly to facebook and there is a whole LINK with a frame and title and it looks better, people engage. How do I get the real deal that shows more of my site name and meta description when sharing?

This is what I want.

THIS IS WHAT I GET (What I do not want)

I would like the content to have my brand name on it. Is this possible?

The page I need help with: [log in to see the link]

 

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