Can Idioms be analyzed grammatically?

[Nh13]

I've started analyzing some texts and then I came to notice that idioms are extremelly difficult to analyze. My question is... Do I analyze the idiom by its constituents parts or as an entire grammatical unit?

For example:

Three 26-year-olds made up their minds that it was time they either left their well-paid jobs in management consultancy and advertising and went into business together or stopped talking about it.

Here, I analyze the idiom made up their minds that it was... as

Verb
made up
Verb Phrase (phrasal verb: lexical verb + adverbial particle)

Direct Object
their minds
Noun Phrase (possessive determiner + noun head)

Subordinate clause...
that is was time...

I looked the idiom up in a Longman dictionary and it also said that the idiom may include 'that'. So I wonder if 'made up their minds that' is a verb prhase and 'it was time...' is the direct object. I'm very confused.

Tell me which Grammar books should I take a look on, too. I'm very interested in sentence analysis (I don't know if this is the right term).
Thanks in advance.

[Tdol]

Idioms are about meaning, so I can't see why breaking them down in grammatical parts helps much. They create a single meaning and they do it in all sorts of ways.