How do I properly introduce myself using a formal register in Italian?
Differently from English, Italian has a specific form to use when addressing people in a formal way. Generally speaking you have to use the third singular person of a verb (Egli-Ella-Lui-Lei) instead of the second (Tu). In this specific example you would specifically say: "Piacere di conoscerla." (Nice to meet you - where "you" is the pronoun used as a form of respect here underlined by the enclitic "-la" instead of a colloquial "you" ideally shown by a "-ti".
To sum up: One has to use "Piacere di conoscerTI" when speaking in an informal occasion and "Piacere di conoscerLA" when addressing someone in a formal way.
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