Twin jets climb fast because of the single engine performance requirement.
+6000fpm is great but you pay awfully heavily for the capability, through carrying a second engine guzzling avtur at some 3 figure # of gallons per hour.
The real trick IMHO would be a < 2000kg SE jet, with a 25k ceiling, 1500nm range (to zero fuel), and ability to operate from a 700m hard runway.
I think a turboprop will get there first though, because of the propwash lowering the stall speed and making the takeoff distance a lot shorter.