"The breakfast's menu" is wrong, the only valid English answer is "The breakfast menu". The menu doesn't belong to breakfast, it belongs to the restaurant and describes the choices available for breakfast.
"Le menu de petit-déjeuner" is not correct in French.
For this sentence "du" is the contraction of "de + le" (literally "The menu of the breakfast").
I guess I'm just confused on why that is. I'll look up the difference between the two.
You may want to take a look at these sources:
Doesn't the first link contradict the answer to this question?
"2. To describe a noun with the de + descriptive noun construction, use de."
Le menu du petit-déjeuner is describing the type of menu isn't it? I'm sure I'm wrong here but curious to know why.
Because here, "du" is a definite contracted article (= de + le) that means "of the".
Though "The menu of breakfast" is wrong, The menu for breakfast" is correct. Why? Both of them don't include "the".
This is a known bug:
- the noun "petit déjeuner" (= breakfast) should not be hyphenated. We'll fix this soon.
- the verb "petit-déjeuner" (= to have breakfast) is hyphenated.