How do you calculate the cost basis of matching shares awarded after a vesting period and then sold under an ESPP program?

[sobermeiert]

I did not receive a 1099-B nor any other documentation other than a Periodic Statement of purchases and sales, since my Employer was Australian.  I looked at my pay slip to find the amount of the ordinary income on my W2 from selling the stock before the two year vesting period.  I do not know what to do with the basis, if any, of the stock that was "Matched" after vesting with no cost to me.

[maglib]

this depends on how they were reported to you and included as w-2 income.  

In the US  old tax laws you were required to pay tax at ordinary income tax rates for the discounted purchase as ordinary income (the discounted cost to you was considered taxable ordinary income).  The free shares would have also been taxed when received at ordinary income tax rates at delivery to you (completely taxed).  LTCG or STCG then would be based on the value of the stock at delivery of both since you already paid ordinary income taxes on the cost of the shares, and decided on if you held them.  

More recently the bargain purchase element was NOT reported until sold.  So to the extent any discount was offered to you for the shares, the bargain purchase element gets reported as ordinary income when sold.
TT will walk you through this reporting if you enter it in ESPP interview.

Read more https://turbotax.intuit.com/tax-tips/investments-and-taxes/employee-stock-purchase-plans/L8NgMFpFX