In the manner or style befitting a doctor; professional and authoritative in the way a physician would be.
From doctor + -ly (from Old English -līc 'having the form or nature of'). The -ly suffix creates adjectives (and later adverbs) from nouns, similar to doctorlike but more archaic in usage.
Though rare today, 'doctorly' shows how English had a whole system of -ly adjectives (scholarly, kingly, manly) before we settled mostly on -like for these meanings—language is always simplifying itself.
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