AFAIK, this is not a standard feature of the original 8051.
Also, I expect that "ONCE" is a trademark of someone or other's specific implementation; therefore you can't expect it to be implemented on any generic processor.

Having said all that, there are some modern 8051 variants which feature on-chip debugging facilities - the Triscend E5 in one.

In the E5, this consists of a hardware breakpoint unit which provides non-intrusive breakpoints on address or data values; ie, they do not affect the processor operation at all.
In addition, the JTAG unit allows you to read and/or modify all register & memory values while the processor is stopped - without the need for any software support (monitors, etc)

I guess that most other processors with some sort of "on-chip debug" would offer similar facilities.

The documentation for your specific processor should tell you what its particular debug facilities can & can't do.