The detection of a high-energy neutrino from the flaring blazar TXS 0506+056 and the subsequent discovery of a neutrino excess from the same direction have strengthened the hypothesis that blazars are cosmic neutrino sources. The lack, however, of $\gamma$-ray flaring activity during the latter period challenges the standard scenario of correlated $\gamma$-ray and high-energy neutrino emission in blazars. We propose instead that TeV-PeV neutrinos are produced in coincidence with X-ray flares that are powered by proton synchrotron radiation. In this case, neutrinos are produced by photomeson interactions of protons with their own synchrotron radiation, while MeV to GeV $\gamma$-rays are the result of synchrotron-dominated electromagnetic cascades developed in the source. Using a time-dependent approach, we find that this "pure hadronic flaring" hypothesis has several interesting consequences. The X-ray flux is a good proxy for the all-flavor neutrino flux, while certain neutrino-rich X-ray flares may be dark in GeV-TeV $\gamma$-rays. Lastly, hadronic X-ray flares are accompanied by an equally bright MeV component that is detectable by proposed missions like e-ASTROGAM and AMEGO. We then applied this scenario to the extreme blazar 3HSP J095507.9+355101 that has been associated with IceCube-200107A while undergoing an X-ray flare. We showed that the number of muon and antimuon neutrinos above 100 TeV during hadronic flares can be up to $\sim3-10$ times higher than the expected number in standard leptohadronic models. Still, frequent hadronic flaring activity is necessary for explaining the detected neutrino event IceCube-200107A.