Extreme cosmic radiation events occurred in the years 774/5 and 993/4 CE, as revealed by anomalies in the concentration of radiocarbon in known-age tree-rings. Most hypotheses point towards intense solar storms as the cause for these events, although little direct experimental support for this claim has thus far come to light. In this study, we perform very high-precision accelerator mass spectrometry (AMS) measurements on dendrochronological tree-rings spanning the years of the events of interest, as well as the Carrington Event of 1859 CE, which is recognized as an extreme solar storm even though it did not generate an anomalous radiocarbon signature. Our data, comprising 169 new and previously published measurements, appear to delineate the modulation of radiocarbon production due to the Schwabe (11-year) solar cycle. Moreover, they suggest that all three events occurred around the maximum of the solar cycle, adding experimental support for a common solar origin.