I think you need more explanation here. You seem to be saying "my employer simply handed me some stock and the value of this stock was included in Box 1 of my W-2." Where you lose me is the "(i.e. tax has been already paid for the original value of the award) as that's not clear. That term "original value of the award" suggests that the number of shares of stock you put in your brokerage account was a subset of the gross number of shares in the award but some shares were withheld "for taxes" and those taxes, too, were reported on the W-2.
If I've got this all right then you can simply tell TurboTax you "purchased" the stock. Your purchase date is the same date that you took possession of the stock and were free to sell them, if you wished. Your per share "purchase price" is the same per share "fair market value" used by your employer to calculate the compensation figure. Generally nothing in your income tax return tells the IRS how you acquired any security you've sold.