  Realization of topological superconductors is one of the most important goals
in studies of topological phases in quantum materials. In this work, we
theoretically propose a novel way to attain topological superconductors with
non-trivial Fermi surfaces of Bogoliubov quasiparticles. Considering the
interacting Luttinger model with $j\!=\!3/2$ electrons, we investigate the
dominant superconducting channels for a multi-orbital quadratic band-touching
system with finite chemical potential, which breaks the particle-hole symmetry
in the normal state. Notably, while the system generally favors d-wave pairing,
the absence of the particle-hole symmetry necessarily induces parasitic s-wave
pairing. Based on the Landau theory with $SO(3)$ symmetry, we demonstrate that
two kinds of topological superconductors are energetically favored; uniaxial
nematic phase with parasitic $s$ wave pairing ($d_{(3z^2-r^2)}\!+\!s$) and
time-reversal-symmetry broken phase with parasitic $s$ wave pairing
($d_{(3z^2-r^2,xy)}\!+\!id_{x^2-y^2}\!+\!s$). These superconductors contain
either nodal lines or Fermi pockets of gapless Bogoliubov quasiparticles and
moreover exhibit topological winding numbers $\pm2$, leading to non-trivial
surface states such as drumhead-like surface states or Fermi arcs. We discuss
applications of our theory to relevant families of materials, especially
half-heusler compound YPtBi, and suggest possible future experiments.
