The idea of the root *h1eg^ ("I", "to speak")

Looking in the new book by Jim P. Mallory and Douglas Adams, I moticed
that there are two identical roots, one meaning "I" (first person
singular nominative) and one meaning "to speak", both with the form
*h1eg^-.

Could this mean that the word for "I" grew out of an expression
meaning "the speaker", as a roundabout alternative to a form beginning
with "m"?
Or is this word meaning "to speak" originally from something like "to
extend oneself"?