I presume it's OK to discuss the answers in more detail now we're past the posting date.
Is this just me? I was expecting the pattern of the dominos or blanks to follow some pattern - anyone have any ideas there? The title is no clue. It looks like some sort of animal - perhaps a dragon with its feet facing right and head facing left - the 2 floating dominos at top left could be the flames? Rotating the grid does not help.
One of the toughest ever I think. Took a long time to get started and once you get the 1st numbers, you think the rest will be easy but you keep getting stuck again. I did use a spreadsheet, not as a brute force cheat but just to save repeating the same calculation over and over when I'd narrowed things down as far as my logic and maths could get. Even when down to the last 2 letters, I and Q were the toughest to get as you had quite a lot of work to find the last domino (2-5), and even then it could go either way - up or down - especially as the shape did not help. I look forward to seeing the purely logical steps on the Listener web site.