Since beskar is an extremely heat-tolerant metal and even survives lightsaber and blaster attacks, how do the Mandalorian metalsmiths forge such a metal?
The only thing that could vaporise the metal seems to be an 'Arc Pulse Generator'.
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We see some beskar being forged in The Mandalorian Season 1, Episode 1 (and then again Episode 4). In short, it's placed on a black metal or ceramic plate that is sitting over a series of blue-flamed burners. The forger presses a button which causes the metal to melt into a liquid which is then poured into a mold. The forger presses another button which stamps the liquid beskar plate into shape and then uses hand-tools to engrave shapes and markings onto it. It seems to be initially workable in much the same way you'd work a soft metal like tin or pewter but then hardens to a material that is stronger than classic steel after it has cooled.
It's not really clear what the button push is doing, possibly passing a current through the metal, creating some kind of induction field (or, more mundanely, maybe just turning the burners up a little).
The concept is simple.
Every metal has a melting point. The Mandalorians just have a forge that can heat the metal to its melting point, which is significantly higher than that of other metals.
As for the Arc Pulse Generator being the only thing that could vaporize it, that's simply not true. The APG never vaporized the metal, it's purpose was to heat the metal until it killed the wearer, not vaporize it. As vaporizing it would be a huge waste of money than if they were to just heat it until the point that they wearers die.
Not only that but in the show The Mandolorian, there is a forge (as shown in one of the previous answers) that can heat the metal to its melting point.
So the conclusion is that they forge it like any other metal, they heat it till it melts, then they use either a casting system or another working system to shape the melted metal to whatever shape they want.
Edit: I did more research and found out that while forging, they don't necessarily temper the Iron, but the Mandalorians actually add other elements to the Beskar Iron to turn it into Beskar Steel which is way more durable. The exact elements and process of adding those elements is a closely-guarded secret of the Mandalorians.