The fastest way to escape kesa gatame is to hook a leg. Once your arm is trapped and the roll is shut down, you expose your opponent's near leg, pull it under you with your own leg, and tip him over the top of your body like a seesaw.
Here's the thing Scott drills into every white belt first. Don't get so focused on the escape that you forget the prevention. Keep that arm in. If he rips it up and out, you are already in the second tier of defenses — and that's where these leg hooks live.
Two options. Same idea.
Option one is the leg hook. If his near knee is hanging out there, weasel your own leg underneath, hook it, then scoot your hips under his base. Once you're underneath him he's teetering — a little shove and he goes. If he's hiding that leg, take your stuck hand and peel the knee out so you can reach it. Expose and hook.
Option two is the head hook. If you can't find the leg, frame his head with your hand — fingertips, knuckle, whatever gets in there — and push it up and away from your chest. That gives you room to swing your leg over the top and hook his head with it. Same finish. Scoot under, tip him over. And it puts serious pressure on his neck if he tries to hang on.
A good judo guy will see this coming and bury his head down low. That's why the frame work matters. Weasel a fingertip in, pop the head up, hook.
For the full picture on fighting out from under a pin, check out our complete guide to escaping side control in BJJ. The complete escape system lives inside Scott Sullivan's BJJ 101 System.