What grammar structure is this? Having finished his reading he went out for a stroll.
She reproach me with having sent her to the Home.
The old fellow fidgeted a bit. Then, “Well, I know I didn’t ought to have done it,”
The "double modal" is considered nonstandard, but it's not uncommon in parts of the American South (and I've been told there are parts of Scotland where it's used too.)
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.