Got to thinking this morning, what was the very first shoulder-fired automatic weapon ? I think the French Chauchat ? Or maybe the German MP18 ? Maybe I'm way off ......

I think it might have been John Browning's conversion of an 1873 Winchester 44-40 carbine to full auto. It's in the Browning museum in Ogden, UT. It was a real stroke of genius. Now it you mean the first full auto that was actually fielded, maybe not commercially successful, I can't help you there.

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The Lewis gun could be fired from the shoulder and first entered service with the Dutch in 1913, followed by adoption by the British in 1915.

It beat the Chauchat to the punch by almost 2 years and the BAR and Johnson 1918 models by almost 5 years.

When the question said "Hand-Held Automatic Weapon" I thought the subject would be automatic weapons designed to be held in one hand. These things do exist. The earliest one I can think of is the Schnellfeuer Broomhandle Mauser. In addition to the automatic fire mechanism, it had a detachable box magazine. While designed for shooting with one hand, it worked a lot better with the shoulder stock.

Girardoni air rifle?
Girardoni air rifle - Wikipedia

Thompson .45 Auto

Depends on how restrictive you want to be regarding the definition. There were late 17th Century examples of man-portable muskets which would fire multiple bullets in succession by using the Roman Candle principle. What we call today automatic weapons just weren’t very practical until the development of metallic cartridges and smokeless powder. John Browning’s conversion of a lever action rifle to muzzle blast actuation was an early example, but I don’t think it was ever fully automatic. There may have been some early adaptations of recoil-operated pistols to full auto fire around the turn of the 20th Century, but I don’t know of any. The Italian Villar-Perosa was an early example of a submachine gun, and I think that was developed around 1914. Another example of a man-portable full auto weapon was the Benet-Mercier automatic rifle which could be carried and fired by one man in much the same way as a BAR could be, but was best operated by a two or three man crew. It dated to around 1908. The Lewis gun is a similar example,as it could be used like a BAR by one man, but was more effective when used by a two or three-man crew.

Stretching the concept a bit might bring in a magazine fed auto loader from third century BC Greece called the Polybolos.



Polybolas - YouTube

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I believe a handful of M1908 Mondragons were built in select-fire (though most were semi-auto).

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where is Ian when you need him...

I believe Steve (I can't remember the model name) had what is considered to be the first submachine gun during ww1, probably around 1916.

Just for the record a photo of the Browning rifle at the museum in Ogden is shown below. According to the placard behind it, the rifle, once activated, “continued firing until the magazine was empty.” So it was automatic in the sense that it used gas from a fired cartridge to run the gun. I suppose you could say it was semi auto in the sense that it still required the trigger be pulled for each shot even though the trigger was mechanically operated by the gas action. Sort of a hybrid.

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The Madsen is just about a hand weapon if you have the muscles. It weighs 20 lbs and dates from 1889 I think. Other than that I'd go with the MP 18 as suggested by the OP.

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Villar Perosa in 9mm caliber. Around WWI time frame. Was also used by gunners in their planes against other nations aircraft. Frank