My youngest son is especially good at it. He can sit perfectly still, arms crossed, with a thousand yard stare for quite a while.
If you haven't already sent him to his room, the best way I've found to beat a pout is with a pout. So, pout back. Mock the hell out of him/her. Cross your arms the same way. Stare at the same nothing. Sit next to him/her and ape their breathing.
One of two things will happen. The kid will crack up and tell you to stop it, dad. Or, the kid will try extra hard to pout. Kids don't hold back. They go full tilt pout right out of the gate. So, when they try to go to hyperpout, they lose their focus and forget why they were pouting. Then you crack.
Okay, there is a possibility that they can pull off the hyperpout. But, they'll stress so much that they'll fart. Then everybody cracks up.