Fat-handle dumbbell lifting is a classic feat of strength of the old-time strongmen. The most famous of these is the 172-pound Thomas Inch dumbbell with its 2 and 3/8ths-inch diameter handle.
There are three lifts that provide just about all you need to take full advantage of the fat handle: the deadlift, clean and press, and snatch.
Naturally, you'll be able to deadlift the most, usually clean and press quite a bit less, and snatch the least. Working on all three lifts will give you several ranges of motion and feelings of acceleration that will improve your thumb and hand strength.