"Predicate" is not a grammatical term; it refers to Logic, not Syntax. There is a predicate in every clause, and normally it's a verb; if it's not, then one needs some form of be to carry the tense in the clause, as in I am tired (where the predicate is tired, a Predicate Adjective) or He is a doctor (where it's doctor, a Predicate Noun).
In the example sentence provided,
- I get the willies when I see closed doors.
the subject of the main clause is I, and so is the subject of the subordinate clause.
The Verb Phrase in the main clause is get the willies, and in the subordinate clause it's see closed doors.
The subordinate clause when I see closed doors is an adverb clause, which modifies the whole main clause and can -- like most adverbs -- appear either at the beginning or the end
of the clause it modifies. If the main clause were longer, it might also appear internally, but this main clause is short and doesn't have any good niches for adverbs.
Adverb clauses are not normally part of either the subject or the verb phrase. This is not true for Noun clauses like Complements, which can be subjects or objects themselves, nor of Adjective clauses like Relatives, which modify and are part of a noun phrase. Noun phrases may be the subject, or may be the object; objects are part of the verb phrase).
I realize that many old steam-powered grammar books talk about the Complete Subject and the Complete Predicate as if they were real constructions that any educated person should know about, but they are one with Phlogisticated Aether and shouldn't be paid attention to. Sorry, things have changed since those books were written.