The 29-year-old was swarmed by fans on the way into the Mandarin Hotel in New York where he signed baseballs, posed for selfies and was super nice to the media.

The questions were rapid-fire ... but here's the gist of it.

-- Cole says signing with the Yanks was a hard decision, which is shocking considered he had 324 MILLION reasons to put pen to paper.

-- Cole explained why he's excited to be a part of one of the most storied franchises in all of pro sports ... "The history. The fans. Competitiveness."

-- And, the beard ... that famous, beautiful beard? He seems to be coming to terms with the fact it's a goner.

We asked Gerrit if he would try and keep his scruff or if he would embrace the Yanks decades-long, clean-cut tradition.

"That's nuts," Cole said ... before saying he would "unfortunately" part ways with his face-warmer. (At least that's what it sounded like he told us. It was super loud.)

The Yankees' courtship of Cole has been one of the biggest stories in baseball ... the guy had a ton of suitors in free agency these past few weeks -- from the Yanks to the Dodgers to the Angels. Everyone was willing to throw insane amounts of cash at him.

Of course, Cole ultimately landed with NY ... and despite him saying it wasn't the easiest choice ever -- he added he's sure as hell happy he made it.