Background: The Doppler broadening of $gamma$-ray peaks due to nuclear recoil from $beta$-delayed nucleon emission can be used to measure the energies of the nucleons. This method has never been tested using $beta$-delayed proton emission or applied to a recoil heavier than $A=10$. Purpose: To test and apply this Doppler broadening method using $gamma$-ray peaks from the $^{26}$P($beta pgamma$)$^{25}$Al decay sequence. Methods: A fast beam of $^{26}$P was implanted into a planar Ge detector, which was used as a $^{26}$P $beta$-decay trigger. The SeGA array of high-purity Ge detectors was used to detect $gamma$ rays from the $^{26}$P($beta pgamma$)$^{25}$Al decay sequence. Results: Radiative Doppler broadening in $beta$-delayed proton-$gamma$ decay was observed for the first time. The Doppler broadening analysis method was verified using the 1613 keV $gamma$-ray line for which the proton energies were previously known. The 1776 keV $gamma$ ray de-exciting the 2720 keV $^{25}$Al level was observed in $^{26}$P($beta pgamma$)$^{25}$Al decay for the first time and used to determine that the center-of-mass energy of the proton emission feeding the 2720-keV level is 5.1 $pm$ 1.0 (stat.) $pm$ 0.6 (syst.) MeV, corresponding to a $^{26}$Si excitation energy of 13.3 $pm$ 1.0 (stat.) $pm$ 0.6 (syst.) MeV for the proton-emitting level. Conclusions: The Doppler broadening method has been demonstrated to provide practical measurements of the energies for $beta$-delayed nucleon emissions populating excited states of nuclear recoils at least as heavy as $A=25$.
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