The answer is me.
In the given sentence, "me" is in objective case of pronoun.
Putting this in sentence pattern/frame perspective:
Subject - Transitive Verb - Indirect Object - Direct Object (S-TV-IO-DO).
Cora - Subject
handed - transitive verb
Tom and me - indirect object
(the pronoun "me" must be used because it is an object pronoun,
while "I" is a subject pronoun.)
applications - direct object.