HMRC trying to charge tax I've already paid!

noh

I do not believe that the US broker, in this case deducts tax and pays it to the IRS or HMRC. My understanding of how it should work is this.

The broker is appointed by the company to administer the share option scheme.
When the option is exercised and the shares are sold, by an employee of the UK subsidary, the broker witholds an amount equal to the tax liability and then passes that amount to the UK subsidary who then pay it to HMRC.
As dori2o says above HMRC will be dealing with your ex employer it is them that provide the figures.
You need to contact your ex employer.
For clarification you could contact the broker and ask them to whom the 40% deducted was paid.

Cook_County

It sounds like you exercised after the P45 was issued so you would have only ever suffered basic rate tax under the PAYE regs at that time.

What does your payslip show?