I don't know of any optimized BLAS library that supports extended precision.
Everything else in the PLASMA software stack can be built from sources,
but if you don't provide fast BLAS, you are not going to get the performance benefit.
This may be a little different for the eigenvalue codes, which depend on BLAS less than other codes,
but in principle this is the story.
That being said, you can try to build PLASMA, and all its software dependencies, using compiler flags for quadruple precision.
That would include compiling Netlib BLAS from sources.
It has never been tried (for PLASMA), but no guts no glory
This is what I would do:
Use PLASMA installer with the --downall option.
This will make sure that all components are built from sources, after downloading from Netlib.
Also, use the --cc option to make sure you are using the Intel compiler,
and use the --ccflags and --fcflags options to compile with extended precision.
The installer should apply those options throughout all software components.
Keep saying Hail Marry throughout the process
Again, this has never been tried, but if you are brave enough, give it a shot.
If this succeeds, you are going to get extended precision LAPACK as a byproduct.
You can compare the performance of LAPACK eigensolver with PLASMA eigensolver.
That would be an interesting experiment.
Let us know how it goes.