We assessed the sensitivity of SKA-like pulsar timing measurements to probe signals of a first order phase transition (FOPT) scenario in the dark sector, which gives rise to primordial black holes (PBHs) and stochastic gravitational waves (GWs). We considered the mechanism where the PBHs are formed from the collapse of Fermi balls created from dark matter filtered out of true vacuum bubbles. In probing these PBHs, the relevant signal is the Doppler phase shift in the pulsar timing signal. Under certain conditions, we found that critical temperatures of interest lie in the 0.1-10 keV range. For the same set of parameters characterizing the FOPT, the resulting stochastic GW spectrum falls within the sensitivity reach of a simultaneous SKA-like search for stochastic GWs, and can be used as a complementary probe. Finally, by considering an explicit form for the dark scalar potential, we performed a parameter scan to identify the class of generic quartic potentials that could lead to viable FOPT scenarios that can be probed by SKA. The results of our scan may provide a useful reference to readily obtain phenomenological constraints on specific models that give rise to FOPTs.