ESP8266 debugger

I am using the ESP8266 platform. I know how to build and load code using the MOS and LOG print.
Beside the load & run method.

Does Mongoose support or provide any debugger (GBD) using VisualStudio or Eclipse ?

I guess you mean GDB, the GNU debugger.
AFAIK the short answer is “I don’t think so”, as that is not likely a mOS issue.
The long answer is:
mOS runs on top of the non-os SDK provided by Espressif, which uses a proprietary event-driven OS.
GDB works as a client-server model, there is a GDB server connected somehow to the device and a client program has a dialog with the server. Typical connections to the device are JTAG and serial port. The device has to have debug support in some form, or the means to implement it in firmware.
Even if all that is available, when you hit stop or when you single step you might find the device inside proprietary code and unless you have the source code for that, you will just see the assembly or the hex. You need an “OS-aware” plugin or IDE.
AFAIK, there is this one guy that developed GDB server support using the serial port, some years ago. Unfortunately I didn’t keep the info around, google it. I honestly don’t think this is useful unless you are developing system code, in this case. I myself am very fond of in-circuit debugging and can barely live without it on microcontrollers where I work at bare metal. Here, working at application level, perhaps the other usual tools will suffice (move a GPIO and check with an oscilloscope, out a code to a group of GPIOs and watch them with a logic-analyzer, log, etc.)

Yes, I found that Arduino has the UART-GDB implemented. This is why I wonder Mongoose available or not.

Thanks for confirmation.