The origin of two large peaks in the atmospheric radiocarbon (14C) 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 14C 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 10Be measurements from both Arctic and Antarctic ice cores. Using ice core 36Cl data in pair with 10Be, we further show that these solar events were characterized by a very hard energy spectrum with high fluxes of solar protons with energy above 100 MeV. 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.