The -ing indicates that "disposing" is a present participle, not a noun. The "of" indicates that it is used as a noun in the greater context of the sentence. Within its own phrase, it is still being used as a verb, though — and can, as such, be modified by an adverb.
This construction is grammatical, idiomatic, and extremely common in English. The term for this is gerund:
In Latin and English grammar, the gerund is a non-finite verb form that can function as a noun. [...]
In English, the gerund is one of the uses of the form of the verb ending in -ing. [...]
A gerund behaves as a verb within a clause (so that it may be modified by an adverb or have an object); but the resulting clause as a whole (sometimes consisting of only one word, the gerund itself) functions as a noun within the larger sentence.
Emphasis added. As you can see, your example sentence is a perfect example of just that.