The sentence you gave does not consist of two subordinate clauses. It contains one independent clause, and one subordinate clause. The internal structure of the sentence goes like this:
[He only gave me [what he owed me.]]
The outer pair of brackets encloses the entire sentence, which is the independent clause. The inner pair of brackets indicates the inner clause. Clauses which are contained within other clauses are known as dependent clauses, and this particular one is a nominal relative clause. It is a relative clause because it begins with the relative pronoun what, and it is a nominal clause (or noun clause) because it functions as a noun within the sentence.
Your intuition is mostly correct, but you've misunderstood where to put the clause boundaries. You seem to have been misled by the false assumption that a clause must be a complete sentence, and the idea that a clause cannot contain another clause. In this case the dependent clause what he owed me is incomplete because relative clauses have to be embedded in a larger context to have meaning, which is why they're called "dependent". And the fragment He only gave me is not even a full clause, because it lacks the direct object that's required by the verb gave. It only becomes a clause when you include the noun clause that acts as its object.
EDIT:
There seems to be some confusion about whether a dependent clause goes inside or outside of the independent clause. Let's look at this deductively, beginning with a simple sentence.
(Abbreviations: [] = clause boundaries, {} = phrase boundaries, IO = indirect object, DO = direct object)
[He gave IO{me} DO{ten dollars}].
In this case, I hope that there is no doubt that the indirect object and the direct object go inside the clause that contains them. The independent clause is not just the subject and the verb, but the subject, the verb, and all of the objects of the verb.
The important thing to remember about noun clauses (and other kinds of subordinate clauses) is that the structure of the independent clause does not change when you insert a noun clause. So in the original example we have something like this:
[He gave IO{me} DO[what he owed me]].
The noun clause what he owed me is the direct object of the verb gave, and it replaces the noun phrase ten dollars. But this has no effect at all on the structure of the independent clause. You can do the same thing with the indirect object:
[He gave IO[whoever he had borrowed from] DO[what he owed them]].
You could go even further with this, adding more nested dependent clauses inside dependent clauses, doing this forever in theory. (In practice it becomes extremely hard to understand after you've nested your clauses more than two or three layers.) But no matter how deep your nesting goes or how complicated the dependent clause becomes, it's still a single component in the structure of the higher-level clause. Dependent clauses do not magically move outside the structure of their parent clauses, nor do they change the grammatical analysis of the clauses that contain them.
The swung dash is most often used to omit something obvious from context for the sake of space. A common use as such is in dictionaries where it can stand for the word currently being defined, though it can also be used as a more general "ditto" mark, to mark information that is omitted or unavailable in transcribing a damaged text, and so on.
As pointed out in another answer, Japanese uses wave dashes (〜) in ways that are analogous to some uses of other dashes in English, such as how the en-dash (–) is used for ranges (1m-2m in English, 1m〜2m in Japanese) and this particular case can sometimes be seen on specification labels on equipment as it stands out a bit more.
That said, it isn't used so much for parenthetical uses of the dash, and the wave dash (〜) is not the same as the swung dash (⁓), so this seems doubtful, though not impossible; borrowings from other cultures and languages aren't always faithful to the source, and certainly uses of punctuation in one language has influenced its use in English whether as a widespread use (e.g. the apostrophe comes from French use, and is now fully a part of English punctuation including in many cases where it does not match French use) or more restricted to a particular community (e.g. the fondness many early-mid 20th Century modernist writers in English, particularly Joyce who insisted adamantly upon them, of the use of em-dashes as speech markers owed a lot to French use—no doubt at least partly related to the number of said writers who lived in Paris).
In all, I can't say this is a standard use, and I can't recommend it: As someone who is used to the swung dash the examples in the question read to me as confusing nonsense because I can't figure out what the words omitted are. That is, I read:
*You are the friend ~ the only friend ~ who offered to help me.
As
You are the friend [text left out] the only friend [text left out] who offered to help me.
And I can't see what that left out text is likely to be here.
Edit:
stib points out that a reasonable reading if one was familiar enough with this cartoon to have remembered it, is:
You are the friendbeast the only friendbeast who offered help to me.
It even explains the lack of commas. The further that is from what was intended (including the missing commas), the more inadvisable it is.
Best Answer
There is one independent clause
followed by a nominative absolute
So called because it describes the action of the subject (nominative) while being syntactically isolated from it (absolute).
The reason that it sounds wrong is because we have no reference for "the glass" other than the display itself. And how can she have her nosed pressed against glass that she hasn't reached yet?