We propose the inverse design of ultracompact, broadband focusing spectrometers based on adaptive diffractive optical networks (a-DONs). Specifically, we introduce and characterize two-layer diffractive devices with engineered angular dispersion that focus and steer broadband incident radiation along predefined focal trajectories with the desired bandwidth and nanometer spectral resolution. Moreover, we systematically study the focusing efficiency of two-layer devices with side length L=100μ m and focal length f=300μ m across the visible spectrum and demonstrate accurate reconstruction of the emission spectrum from a commercial superluminescent diode. The proposed a-DONs design method extends the capabilities of efficient multi-focal diffractive optical devices to include single-shot focusing spectrometers with customized focal trajectories for applications to ultracompact spectroscopic imaging and lensless microscopy.