All the Kings' Men

With a heavy sigh the boy at his back collapsed, bright green eyes flicking over his shoulder as All Might lost the battle to maintain his form...but that was alright. He recognized the scrawny, bruised boy staring up at him. A wall grew up to separate them from the students, but the boy didn’t seem to notice.
“Are you alright, All Might?”
The boy had been skinny when he’d last seen him, but now there was a slightly gaunt look to his bruised face and a scar like peeling paint stretched up from his jaw. The mess of dark curly hair was the same though, as was the light in those shy eyes. This boy…where had he even come from? Exhausted, Toshinori allowed himself to drop into the dirt beside him.
“Thanks to you…Midoriya Izuku.”
One of Midoriya’s eyes was swelling shut, but the other still widened in shock before a blazingly bright smile broke across his face.
 
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TLDR: Midoriya is kidnapped by Shigaraki for a year and is slowly adopted by 1-A as he works through the resulting trauma and fights to become a hero.