No matter which 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."

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."