This is a totally different technology.
A free electron laser (FEL) uses free electrons (electrons not attached to a nucleus) as a lasing medium to produce light. The light would shine through a mask and expose photoresist more or less just like the light from ASML’s tin plasma contraption, minus the tin plasma. FELs, in principle, can produce light over a very wide range of wavelengths, including EUV and even shorter.
That DARPA thing is a maskless electron beam lithography system: the photoresist is exposed by hitting it directly with electrons.
Electrons have lots of advantages: they have mass, so much less kinetic energy is needed to achieve short wavelengths. They have charge, so they can be accelerated electrically and they can be steered electrically or magnetically. And there are quite a few maskless designs, which saves the enormous expense of producing a mask. (And maskless lithography would let a factory make chips that are different in different wafers, which no one currently does. And you need a maskless technique to make masks in the first place.) There were direct-write electron-beam research fabs, making actual chips, with resolution comparable to or better than the current generation of ASML gear, 20-30 years ago, built at costs that were accessible to research universities.
But electrons have a huge, enormous disadvantage: because they are charged, they repel each other. So a bright electron beam naturally spreads out, and multiple parallel beams will deflect each other. And electrons will get stuck in electrically nonconductive photoresists, causing the photoresist to (hopefully temporarily) build up a surface charge, interfering with future electron beams.
All of that causes e-beam lithography to be slow. Which is why those research fabs from the nineties weren’t mass-producing supercomputers.