  The Hilda asteroids are among the least studied populations in the asteroid
belt, despite their potential importance as markers of Jupiter's migration in
the early Solar system. We present new mid-infrared observations of two notable
Hildas, (1162) Larissa and (1911) Schubart, obtained using the Faint Object
infraRed CAmera for the SOFIA Telescope (FORCAST), and use these to
characterise their thermal inertia and physical properties. For (1162) Larissa,
we obtain an effective diameter of \textcolor{black}{46.5$^{+2.3}_{-1.7}$~km,
an albedo of 0.12~$\pm$~0.02, and a thermal inertia of 15$^{+10}_{-8}$
Jm$^{-2}$s$^{1/2}$K$^{-1}$. In addition, our Larissa thermal measurements are
well matched with an ellipsoidal shape with an axis ratio a/b=1.2 for the
most-likely spin properties. Our modelling of (1911) Schubart is not as
refined, but the thermal data point towards a high-obliquity spin-pole, with a
best-fit a/b=1.3 ellipsoidal shape. This spin-shape solution is yielding a
diameter of 72$^{+3}_{-4}$ km, an albedo of 0.039$\pm$~0.02, and a thermal
inertia below 30 Jm$^{-2}$s$^{1/2}$K$^{-1}$ (or
10$^{+20}_{-5}$Jm$^{-2}$s$^{1/2}$K$^{-1}$).} As with (1162) Larissa, our
results suggest that (1911) Schubart is aspherical, and likely elongated in
shape. Detailed dynamical simulations of the two Hildas reveal that both
exhibit strong dynamical stability, behaviour that suggests that they are
primordial, rather than captured objects. The differences in their albedos,
along with their divergent taxonomical classification, suggests that despite
their common origin, the two have experienced markedly different histories.
