  Using high-resolution simulations of black hole formation from the direct
collapse of massive stars undergoing pulsational pair-instability supernovae
(PPISN), we find a new phenomenon which significantly affects the explosion and
leads to two peaks in the resulting black hole mass function (BHMF). Lighter
stars experiencing the pair-instability can form a narrow shell in which alpha
ladder reactions take place, exacerbating the effect of the PPISN. The shell
temperature in higher mass stars ($>62 {\rm M}_\odot $ at the onset of helium
burning for population-III stars with metallicity $Z=10^{-5}$) is too low for
this to occur. As a result, the spectrum of black holes $M_{\rm BH} (M_i)$
exhibits a shoulder feature whereby a large range of initial masses result in
near-identical black hole masses. PPISN therefore predict two peaks in the mass
function of astrophysical black holes -- one corresponding to the location of
the upper black hole mass gap and a second corresponding to the location of the
shoulder. This shoulder effect may explain the peak at $35_{-2.9}^{+1.7}{\rm
M}_\odot$ in the LIGO/Virgo/KAGRA GWTC-3 catalog of merging binary black holes.
