Portable optical spectrometers are crucial devices for bio-chemical sensing and spectroscopic applications whereby robust, compact and cost-effective set-ups are desirable. However, existing miniaturized instruments typically struggle to achieve broad wavelength operation and high spectral resolution at the same time. Here, an all-fiber optical spectrometer based on two cascaded Bragg gratings is devised and demonstrated, showing a record resolution and a wavelength span-to-resolution ratio larger than that of most miniature broadband spectrometers reported to date. Thanks to a synchronous control of the grating lengths and to a unique combination of their reflection features, spectral analysis of incoherent light within 1 pm is achieved. On the other hand, fast and reproducible wavelength tuning over several nanometers on a millisecond-timescale is ensured by mechanical stretching of the internal fiber, limited only by the actuator's dynamic range. A striking evidence of the spectrometer capabilities is provided with Doppler-limited spectroscopy of gas absorption bands performed with a near-infrared LED source. The observed spectra exhibit lineshapes comparable with those obtained by laser-based set-ups and the retrieved gas-line parameters are in agreement with existing spectroscopic databases. The spectrometer lends itself to applications in high-resolution interrogation of multiple fiber-optic sensors as well as broadband imaging with supercontinuum light.