Pretty sure there is no instance where breakfast can be used as a verb. So to answer your question, it's a prepositional phrase. I guess technically it's grammatically correct, but usually you would say "to eat breakfast" or "for breakfast" instead.

From a dictionary: breakfast: noun 1. the first meal of the day; morning meal: A hearty breakfast was served at 7 a.m. 2. the food eaten at the first meal of the day: a breakfast of bacon and eggs. verb (used without object) 3. to eat breakfast: He breakfasted on bacon and eggs. verb (used with object) 4. to supply (someone) with breakfast: We breakfasted the author in the finest restaurant. James' answer seems to be the most accurate. In Canada and the US we use breakfast as a noun - maybe 99% of the time (definition #1 and 2) It is very unusual for someone to use breakfast as a verb, although it's not incorrect (definition #3 and 4). "and sat down to breakfast" means "and sat down to eat the breakfast". "to breakfast" would usually be interpretted as a preposition + noun, however, it could also be considered to be the infinitive form of the verb breakfast (i.e., to breakfast).

To break-fast, it can be used as a verb ,it means you haven't eating for sometime and it's time to break the fasting . (It's time to break-fast ,i have been fasting for 10 hours.) now of course breakfast is a noun .( I have breakfast a 7:00 o'clock in the morning.)

In your example, it could be either a prep+ noun or a verb, but this is probably a prep + noun. Generally, "breakfast" is used as a noun, but it does depend on the context. For example: "Let's go to breakfast at 10 o'clock" (prep + noun) "Let's breakfast at 10 o'clock" (verb)

Here to breakfast is a verb other than a noun.