The origin of two large peaks in the atmospheric radiocarbon (C-14) concentration at AD 774/5 and 993/4 is still debated. There is consensus, however, that these features can only be explained by an increase in the atmospheric C-14 production rate due to an extraterrestrial event. Here we provide evidence that these peaks were most likely produced by extreme solar events, based on several new annually resolved Be-10 measurements from both Arctic and Antarctic ice cores. Using ice core Cl-36 data in pair with Be-10, we further show that these solar events were characterized by a very hard energy spectrum with high fluxes of solar protons with energy above 100MeV. These results imply that the larger of the two events (AD 774/5) was at least five times stronger than any instrumentally recorded solar event. Our findings highlight the importance of studying the possibility of severe solar energetic particle events.