Phonological reconstruction is one of the central problems in historical linguistics where a proto-word of an ancestral language is determined from the observed cognate words of daughter languages. Computational approaches to historical linguistics attempt to automate the task by learning models on available linguistic data. Several ideas and techniques drawn from computational biology have been successfully applied in this area of computational historical linguistics. Following these lines, we adapt MSA Transformer, a protein language model, to the problem of automated phonological reconstruction. MSA Transformer trains on multiple sequence alignments as input and is, thus, apt for application on aligned cognate words. We, hence, name our model as Cognate Transformer. We also apply the model on another associated task, namely, cognate reflex prediction where a reflex word in a daughter language is predicted based on cognate words from other daughter languages. We show that our model outperforms the existing models on both the tasks, especially when it is pre-trained on masked word prediction task.