So, after I found about UniversalPauseButton, Googling for this ("windows SIGSTOP"), getting this question as the first search result (thanks Ilia K. your comment did its job), and reading the answers, I went back to checkout the code.
Apparently, it uses undocumented NT kernel and Win32 APIs _NtSuspendProcess, _NtResumeProcess and _HungWindowFromGhostWindow.
PsSuspend, the utility you mentioned and linked to probably uses these APIs, I couldn't verify this, the source code isn't supplied, only executables and a EULA, you can probably figure that out by disassembling the binary but it's against the EULA.
so, to answer your specific question, checkout UniversalPauseButton's main.cpp, basically you call _NtSuspendProcess(ProcessHandle) and _NtResumeProcess(ProcessHandle), ProcessHandle being the handle of the process you want to pause or resume.