How do I take apart Pegasus handle on model #856 0111bn? Handle leaks from the back of the middle joint.

Replacing Cartridge On Pegasus Kitchen Sink Handle - Plumbing - DIY Home Improvement | DIYChatroom

Mostly all kitchen faucet handles are held on by an allen set screw unless there is a screw under an insignia cover. Almost sure you have an allen set screw though.

Check out these videos for a guide to how it goes. Listen carefully to the guy on the second video explaining how to get a free cartridge replacement.

https://www.youtube.com/watch?v=U7PtDhV30E4

I have noticed most do, this particular model does not.

Show us a photo of your faucet.

Does it look like this ? If so, the button on top unscrews ....

Milo

So how do you get the button to unscrew? Cannot get it to unscrew and mine has the same issue of leaking. Also, any idea what the cartridge number is for replacement. Additionally, I used the sprayer once and it blew out and is leaking at the connection now. Any suggestions? I JUST installed this although we bought it YEARS ago for a different house and never installed it.

You may (or may not) be able to grip the edges of the button with a pair of needle nose pliers.
You could also try putting a small piece of double sided tape on the button, then pressing down with your thumb and twisting, get the button to unscrew.

Do as Harold said.
Also, that button may be just pushed in. It is held in by O-ring. Take some sharp object, insert it into the seem between the body of the faucet and the nut, and see if you can carefully pry it off

Milo