Colloidal semiconductor nanoplatelets exhibit exceptionally narrow photoluminescence spectra. This occurs because samples can be synthesized in which all nanoplatelets share the same atomic-scale thickness. As this dimension sets the emission wavelength, inhomogeneous linewidth broadening due to size variation, which is always present in samples of quasi-spherical nanocrystals (quantum dots), is essentially eliminated. Nanoplatelets thus offer improved, spectrally pure emitters for various applications. Unfortunately, due to their non-equilibrium shape, nanoplatelets also suffer from low photo-, chemical, and thermal stability, which limits their use. Moreover, their poor stability hampers the development of efficient synthesis protocols for adding high-quality protective inorganic shells, a well-known strategy to improve the performance of quantum dots. Herein, we report a general synthesis approach to highly emissive and stable core/shell nanoplatelets with various shell compositions, including CdSe/ZnS, CdSe/CdS/ZnS, CdSe/CdxZn1-xS, and CdSe/ZnSe. Motivated by previous work on quantum dots, we find that slow, high-temperature growth of shells containing a compositional gradient reduces strain-induced crystal defects and minimizes the emission linewidth while maintaining good surface passivation and nanocrystal uniformity. Indeed, our best core/shell nanoplatelets (CdSe/CdxZn1-xS) show photoluminescence quantum yields of 90% with linewidths as low as 56 meV (19.5 nm at 655 nm). To confirm the high quality of our different core/shell nanoplatelets for a specific application, we demonstrate their use as gain media in low-threshold ring lasers. More generally, the ability of our synthesis protocol to engineer high-quality shells can help further improve nanoplatelets for optoelectronic devices.