Which turboTax product do I need to produce 1 1099-NEC to a hired help? Do I need to wait after my turbotax filing to issue (send a copy) that 1099-NEC?

No, in fact you should file any 1099s that you need to file BEFORE you file your tax return.  And the sooner the better because they were all supposed to have been filed by January 31st.

Here's where to file 1099s using TurboTax.

@kingsingchow 

You can prepare W2 and 1099NEC & 1099MISC in either the Desktop Home & Business program or the Online Premium version using the QEF Quick Employer Forms.

How to prepare 1099 & W2
https://ttlc.intuit.com/community/forms/help/how-do-i-create-w-2-and-1099-forms-in-turbotax/00/25869