Quantum information science promises transformative impact over a range of key technologies in computing, communication, and sensing. A prominent example uses entangled photons to overcome the resolution-degrading effects of dispersion in the medical-imaging technology, optical coherence tomography. The quantum solution introduces new challenges: inherently low signal and artifacts, additional unwanted signal features. It has recently been shown that entanglement is not a requirement for automatic dispersion cancellation. Such classical techniques could solve the low-signal problem, however they all still suffer from artifacts. Here, we introduce a method of chirped-pulse interferometry based on shaped laser pulses, and use it to produce artifact-free, high-resolution, dispersion-cancelled images of the internal structure of a biological sample. Our work fulfills one of the promises of quantum technologies: automatic-dispersion-cancellation interferometry in biomedical imaging. It also shows how subtle differences between a quantum technique and its classical analogue may have unforeseen, yet beneficial, consequences.