We study a two-level impurity coupled locally to a quantum gas on an optical lattice. For state-dependent interactions between the impurity and the gas, we show that its evolution encodes information on the local excitation spectrum of the gas at the coupling site. Based on this, we design a nondestructive method to probe the system's excitations in a broad range of energies by measuring the state of the probe using standard atom optics methods. We illustrate our findings with numerical simulations for quantum lattice systems, including realistic dephasing noise on the quantum probe, and discuss practical limits on the probe dephasing rate to fully resolve both regular and chaotic spectra.