How would this freezing leave a "trace"?

Frost leaves a record by damaging the cell structure, creating a "frost ring". Hadd et al discuss data on frost rings in N. American trees.

There is specific discussion of the Fimbul event (in AD 536) in this paper by Helama et al. 2019 (unfortunately with a pay wall). There is also discussion that there may have been a similar event in 1627 BC.

If the data is good enough, it may be possible to get an estimate of the time of year at which the damage occurred by looking at the structure of the damaged cells and seeing how it fits with what is known about the annual cycle of cell growth. This requires more detailed analysis than the widely used approach of measuring ring thickness to estimate annual mean (or growing season mean) temperature.

There is more recent information on ultra-cold summers in the Northern hemisphere, the "year without winter".