A microkernel fully hardcoded in hardware?
The key search terms here are FPGA and ASIC, since at this point the OS is no longer running on instructions executed by a CPU in the conventional sense.

Here is a relevant result I found with a quick Google search on microkernel fpga and os kernel fpga:

Susanna Nordström et al. Application Specific Real-Time Microkernel in Hardware.


  In the RTU, the scheduling, inter process communication, interrupt management, resource management, synchronization and time management control are implemented in hardware (using VHDL).


This paper goes a step further and describes an OS-like environment for FPGA programs themselves: Kermin Fleming et al. The LEAP FPGA Operating System.


  Like a software operating environment, LEAP provides both basic device abstractions for FPGAs and a collection of standard I/O and memory management services.