Micrometer-sized Janus particles of many kinds can be formed using droplet microfluidics, but in existing methods, the microfluidic templating is strongly coupled to the material synthesis, since droplet solidification occurs through rapid polymerization right after droplet formation. This circumstance limits independent control of the material properties and the morphology of the resultant particles. In this paper, we demonstrate a microfluidic technique to produce functional Janus microgels from prefabricated, cross-linkable precursor polymers. This approach separates the polymer synthesis from the particle gelation, thus allowing the microfluidic droplet templating and the functionalization of the matrix polymer to be performed and controlled in two independent steps. We use microfluidic devices to emulsify semidilute solutions of cross-linkable, chemically modified or unmodified poly(N-isopropylacrylamide) precursors and solidify the drops via polymer-analogous gelation. The resultant microgel particles exhibit two distinguishable halves which contain most of the modified precursors, and the unmodified matrix polymer separates these materials. The spatial distribution of the modified precursors across the particles can be controlled by the flow rates during the microfluidic experiments. We also form hollow microcapsules with two different sides (Janus shells) using double emulsion droplets as templates, and we produce Janus microgels that are loaded with a ferromagnetic additive which allows remote actuation of the microgels.