Grammatically these 2 sentences seem to have the same structure
Why is it that "I am finished my sandwich" sounds correct but "I am started my sandwich" does not?
Whether either of these sentences sounds correct, and if so, which of them, actually depends on your dialect. There's a good post about this on the website of the Yale Grammatical Diversity Project.
(Here's a sample map from them of the US -- they have a much better, interactive version in the article, but I thought I'd include this screenshot here just to have a visual. The redder a state is, the greater proportion of people there are who judged "I'm done my homework" as acceptable; the number represents the total amount of people who participated from the state.)
We can also consider a few more sentences:
"I'm finished" and "I'm done." Before today, I assumed these sentences would be generally accepted as grammatical by speakers of all dialects of English, but apparently "I'm done" has historically been more common in North America than in Britain. It does appear that these usages crossed the Atlantic at some point, and nowadays at least some British English speakers find them acceptable.
For many speakers (myself among them), "am finished" and "am done" can only be used intransitively, which means there is no direct object. For these speakers, sentences like "I am finished my sandwich" and "I am started my sandwich", with a direct object "my sandwich," sound incorrect.
But for some North American English speakers (apparently speakers of Canadian English in particular, but also a minority of speakers in the United States), "am done" and "am finished" can be used transitively. If the above sentences with "sandwich" sound correct to you and you're a native English speaker, you belong to this group.
For a subset of these speakers, "am started" can also be used (transitively, and I'd assume also intransitively).
Relevant quotes:
Who says this?
The done my homework construction is a widespread characteristic of Canadian English, and it is also found in the United States among speakers in Philadelphia, Vermont, and New Hampshire (Yerastov 2008, 2010a, 2010b, 2012, submitted; Hinnell 2012; Fruehwald and Myler 2013, 2014). It has not been found in the dialects of the United Kingdom or elsewhere outside of North America.
In particular see
Allowed verbs
There is variation across dialects in terms of which verbs speakers allow in this construction. According to Yerastov (e.g. 2010a, 2010b), there is a hierarchy along the lines of finished > done > started. In other words, if speakers accept started (as in I'm started my homework), they will accept all three verbs. If speakers accept done, they will also accept finished, but not necessarily started. Finally, some speakers accept only finished. This kind of hierarchy resembles the one found in the needs washed construction with need > want > like (see here for further discussion of the needs washed construction).
No matter which way you look at "am finished," as a passive construction or a verb plus a complement, there is no place for a direct object. If you want to maintain the verb, you have to say
I am finished with my sandwich
and if you want to maintain the direct object, you need a transitive and active verb:
I finished my sandwich.
"I am finished" is idiomatic for "I am done," but "I am started" isn't idiomatic for "I have begun."
Neither I am finished my sandwich nor I am started my sandwich is grammatical.
Am can only appear as a verb in a set number of ways:
The verbs start and finish form a contrasting pair: to indicate that some task has begun and that some task has ended. But you are not the one being started or finished; it is the sandwich, or it is the task of making, packaging, eating, or whatever it is you are doing to the sandwich that is being started and finished.
The progressive/continuous aspect is not in use, so am should indicate passive voice or a copula. Since finish and start are monotransitive, their passive forms do not have direct objects, and agency would be indicated with a prepositional phrase headed by by. But you are not trying to say I am finished by my sandwich or I am started by my sandwich, rather the other way around.
The am in I am finished with my sandwich is a copula, with finished being a predicate adjective, defined by ODO as
1b. In a state of having completed something
and sandwich being the object of the preposition with.
Started lacks this adjectival use. Neither AHD nor ODO nor MW nor LDOCE gives it a separate entry, and you would have a hard time finding native speakers who say I am started in contrast with I am finished, which is idiomatic to announce the completion of any meal or task.
Depending on what you want to express, you would instead say I finished my sandwich, or I have finished my sandwich I am finished with my sandwich, or I started [with/on/at] my sandwich, or I am starting on my sandwich, or I have started [with/on/at] my sandwich. Adding an elucidating infinitive or gerund complement would be even better: I finished eating my sandwich, I started to make my sandwich.
To me "I am finished my sandwich" is not correct. We say, "I have finished.... But who am I to say that what you say is not correct. It may have been at one time or it may be in the future. So language does change and if somebody wants to stay "I am started my sandwich" then so be it. It sounds odd to me but I understand what is meant and as long is what a person says in the ordinary course of events in comprehensible, that is fine by me. HOWEVER in some circumstances (say) a scientific or political discussion I hope that what a person says is comprehensible to everybody concerned. If not a whole lot of time can be wasted sorting things out.
They both make sense, but you rarely will hear "I started (eating) my sandwich" because the person's mouth is full of sandwich. It's not until they are finished that they can say, with an empty mouth "I finished (eating) my sandwich"
