The exploration of atomic fractional quantum Hall (FQH) states is now within reach in opticallattice experiments. While bulk signatures have been observed in a system realizing the Hofstadter-Bose-Hubbard model in a box [Leonard et al., arXiv:2210.10919], how to access hallmark edge properties in this setting remains a central open question. We propose and analyze a realistic scheme to extract the momentum-resolved edge spectrum of atomic FQH states. Our proposal is based on subjecting the prepared FQH ground state to two interfering Laguerre-Gaussian beams, which transfer a controlled angular momentum l and energy ℏω to the system. The resonant coupling is then detected through local density measurements, by tracking the transfer of atoms from the bulk to the edge of the FQH droplet. We benchmark our method using numerical simulations of the Hofstadter-Bose-Hubbard model, considering few bosons in the ν = 1/2 Laughlin ground state. These studies demonstrate that our probing scheme is well suited to extract hallmark features of FQH spectra: a chiral gapless edge branch and a gapped magneto-roton mode. These signatures are already detectable in realistic systems of two bosons, provided that the box potential is larger than the droplet. Our work paves the way for the detection of fractional statistics in cold atoms through edge signatures.