Lunch for Wesley meant "eating a peanut butter sandwich". What kind of phrase are the bolded words?

It above phrase is a verb phrase in present continuous tense "He is eating"

This is a verb phrase because the head of the verb phrase (VP) is the word "eating" (verb).

To understand the anatomy of the sentence. You must know why. Sentences have different tiny phrases to make larger phrases to make a whole sentence. The VP (verb phrase) also holds a determiner phrase (DP) inside. What is he eating? (A peanut butter sandwich) <= = Determiner phrase because of the word "a". The DP has a noun phrase (NP) inside. The Peanut butter sandwich" is the noun phrase or NP