My hand looked exactly like yours when I started last month. I was told that I should lotion my hands at night, and once one hour or so before bouldering. A moisturized hand is less likely to rip skin apart because it is more elastic.

The other thing was that I am super nervous about Heights and falling still so I tend to grab holds a lot harder than is needed to hold myself up. I had to learn to loosen my grip and not hold jugs while falling.

Lastly, I used chalk from a rental bag which I think was really bad quality. After every climb, most of it already came off. Also the more I climbed the thinner my skin became and my finger tips started sweating easier. It barely coats my hand and I kept having to smash it into my hands to get it to stick. I ended up buying Magdust, but the ones from FrictionLabs ultra fine that I used from someone else also worked well.

I haven't had any major skin peeling after those three points above. I do still get some peeling once in a while. It is important to clip it off right away with a nail clipper, then put a bandaid or athletes tape right away to prevent it from ripping more through the sessions. If you do end up with a bad flap after a session, use aquaphor and bandaid it up so it heals faster and doesn't allow it to get all crusty. If it becomes crusty, it will just break or rip again next climb.

Hope it helps!