In the sentence "He was, in the words of one fan, 'eliminating obesity one Netflix binge at a time.'", the phrase "one Netflix binge at a time" serves the following grammatical roles:

1. Adverbial Phrase:

• The phrase "one Netflix binge at a time" functions as an adverbial phrase, modifying the verb "eliminating."
• It tells how the action of "eliminating obesity" is supposedly being carried out—gradually, step by step, with each "Netflix binge."

2. Prepositional Phrase:

• Within this adverbial phrase, "at a time" is a prepositional phrase with "at" as the preposition and "a time" as the object of the preposition. This part of the phrase expresses the rate or manner of the action.

3. Noun Phrase:

• "One Netflix binge" is a noun phrase that acts as part of the adverbial expression. It specifies the unit by which the action is progressing, with "Netflix binge" as the noun modified by "one."

Summary:

The phrase "one Netflix binge at a time" modifies the verb "eliminating," explaining how the action is occurring. It functions adverbially to suggest a gradual process or incremental approach, adding a humorous contrast to the idea of eliminating obesity through sedentary activity like binge-watching.