This is not a technical concept or method of linguistics that you can "look up" based on a name. It is related to something that linguists do, but again that practice doesn't have a specific name that you could read up on.
The premise is that speakers of a language can do things with language, and their behavior is informative with respect to grammatical properties. Examples are speech errors (unintentional modifications of language output), language games, rhymes and various kinds of psychological perception experiments (presenting artificial stimuli to see what responses speakers give). These could be grouped together as "metalinguistic tasks", meaning that the subject uses their knowledge of language to do something else.
One somewhat common task of this kind which linguists perform is "syllable counting", where a speaker is asked how syllables there are is such-and-such word. A related task is the syllable-break task, where a person is asked to say a word syllable-by-syllable. Whether or not such data has scientific validity is an open question, especially when the subject isn't a highly educated speaker of English ("What is this word 'syllable' that you keep using?").
What you describe is similar to an artificial language learning experiment. The problem is that this doesn't correspond to anything that English speakers might do on their own, so you have to set up the experiment carefully (don't assume an answer to the question that you are trying to answer). Generally, the procedure is to train subjects on a limited subset of contexts, for example you could present them with (auditory) stimuli like [kæt], [kæʔæʔæ:t] [dɔg], [dɔʔɔʔɔ:g] and so on and then see what they produce with "goat", "bite", "couch", "boil", "chaos". Then there is a further leap that you have to take that relates these results to the concept "diphthong".
This procedure does, however, presuppose that you can transcribe words phonetically (and think about competing phonological hypotheses). If you have no idea what letters to use in transcribing "mouse", the experiment is probably not going to tell you anything. You have to be able to hyper-accurately transcribe the data produced by speakers, without imposing an invalid theory on the data. IMO, [mæ-æ-æʊs] is an extremely unnatural production, and [mæʊ-æʊ-æʊs] is much more natural (and [mɑ-ɑ-ɑʊs] is completely impossible, though 'fight' → [fɑ-ɑ-ɑɪt] is at least imaginable. You need to be able to detect that the nuclear vowel in the "i" and "ou" diphthongs is a different vowel, when a speaker uses different vowels.