Forgotten Title Transmigration Protagonist is a daughter of the Emperor and she collects her imperial brothers

I remember her being abandoned at the start in the cold palace, she’s very good at gaining favor and even her half sisters( the princess) also likes her. I’m kind of remember her goal is to make sure the brothers didn’t end up killing each other over the throne or something or maybe it’s was just to survive . She befriend this particular brother who’s almost in the same situation as her, he stole a chicken just to give it to her and because MC ‘s mental age is much older, she kind of pitied the poor young boy.

 

I'm not sure if this is what you're looking for but coincidentally i'm reading this book and some of them are same with your description (but remember that I just started this).

Is it, "Rebirth of the Tyrant’s Pet: Regent Prince is too Fierce"?

MC is at the cold palace and this sickly bro also gave her chicken.

 

Villainess Wants To Turn Over A New Leaf maybe???