give your eye teeth for
give (one's) eye teeth for (something)
To give or relinquish anything (or something very valuable) in order to receive some specific thing in return. Oh, I would give my eye teeth for curly hair like yours.
give your eye teeth forgo to any lengths in order to obtain something.
The eye teeth are the two canine teeth in the upper jaw.
1930 W. Somerset Maugham Cakes & Ale He'd give his eye-teeth to have written a book half as good.