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Since Beethoven's early works are considered part of the Classical era, and his later works are considered part of the Romantic era, is there, by scholarly consensus, a single piece of his that indicates the changeover?
I've heard a technical explanation of Romantic harmony as an expansion of Classical harmony to include erstwhile dissonances such as the dominant 9th chord. So I imagine this first piece (if it exists) should introduce the power of the dominant 9.
The defining point lies somewhere between the Third Symphony and the Fifth Symphony.
In particular, I would argue that it's the Fourth Piano Concerto where Beethoven makes the most radical break from Classical to Romantic music, inasmuch as the harmonic freedom exploited in late Mozart and in Beethoven's earlier works is combined with breaking structural "norms"—the soloist comes in before the orchestra for the first time, the middle movement features piano versus orchestra (instead of the "first among equals" model that predominated until that time), and the "rebalancing" of the work making the allegro-sonata opening as long as the remaining movements combined.
In order to answer your question, the question itself needs to be modified. To correct your thought, the Romantic Period did not occur specifically during Beethoven's lifetime, so it could therefore not have happened during his "middle" period.
It is important to understand that when talking about labeling a period of music is to label a zeitgeist of thought, and because of that, it is fairly impossible to distinguish a particular moment or instance in which the shift occurred. That said, with every musical period there are events which reinforce the current zeitgeist. This is consistent throughout history. These events are related to the people that we end up studying in school; such as Beethoven.
Beethoven himself is seen as one of the strongest (and the most famous) connections between the Classical period and the Romantic period. However, it is generally considered that for music, what would become the Romantic period began to seriously gain momentum in the early 1830's, a few years after Beethoven's death in 1827.
It is because of this that there cannot be scholarly consensus on a single piece of music that would influence and create and entirely new musical period. At best, a single piece like that would have been seen as an intriguing outlier and imitated by less famous composers.
Regarding the technical explanation that you have heard, that is just not true. Long before Beethoven's time, Mozart and other composers were using 13th chords, non-functional harmony, chromatic passing tones, and non-local harmonic modulations. Beethoven would often use unresolved, harmonically non-functional fully-diminished chords as a means of sustain tension throughout a passage. In comparison to a dominant 9th chord, it's like whipped cream.
In addition to expanded harmony, the Romantic period was a result of composers playing with conventions - both with formal organization, with listeners' expectations, and the beginning of the composer as a self-developing artist.
At the risk of turning this into a paper, I think I'll stop there.
I hope that answers your question.
One piece which is often mentioned is Beethoven's 3rd symphony. I don't think harmony alone could be a defining factor. Bach already has some pretty wild stuff. There's an extremely dissonant chord-progression piece (or section of a piece) by him, but I don't remember what it is (it's not the chromatic fantasie and fugue).
