Search found 2 matches: Chan-wook Park

Filmmakers with multiple notable films that have started since 2000 that are very good:

Alfredson, Tomas (Let the Right One In, Tinker Tailor Soldier Spy)*
Bong Joon-Ho (Memories of Murder, The Host, Mother)
Castaing-Taylor, Lucien (Leviathan, Sweetgrass)
Dominik, Andrew (The Assassination of Jesse James by the Coward Robert Ford, Killing Them Softly)
Glazer, Johnathan (Under the Skin, Birth)*
Green, David Gordon (George Washington, All the Real Girls)*
Kaufman, Charlie (Synecdoche New York, Anomalisa)
Lanthimos, Giorgos (Dogtooth, Alps, The Lobster)*
Madsen, Michael (Into Eternity, The Visit)*
Oppenheimer, Joshua (The Act of Killing, The Look of Silence)*
Paravel, Verena (Leviathan, Foreign Parts)
Park Chan-wook (Oldboy, Stoker)*
Perry, Alex Ross (The Color Wheel, Listen Up Philip, Queen of Earth)
Strickland, Peter (Berberian Sound Studio, The Duke of Burgundy)
Weerasethakul, Apatchitpong (Tropical Malady, Syndromes and a Century, Uncle Boonmee)*
West, Ti (The House of the Devil, The Innkeepers)
Wright, Edgar (Shaun of the Dead, Hot Fuzz, Scott Pilgrim, The World's End)

* = meets your criteria for me

The 2000 cut-off seems arbitrary, as it ignores people who just started a few years earlier but have produced their best stuff in the 21st century: Paul Thomas Anderson, Spike Jonze, Noah Baumbach, Kim Ki-Duk, Kim Ji-Woon, Wes Anderson, Nicholas Winding Refn, Lynne Ramsay, the Dardennes, Kelly Reichardt, Jafar Panahi, off the top of my head.

If I had to put money down on which will have the most relevance in the future, I'd go Weerasethakul, Kaufman, Perry.

https://www.criticker.com/people/Hogg_Joanna/
https://www.criticker.com/people/Andrea_Arnold/
https://www.criticker.com/people/Duane_Hopkins/

some more names for your scoff and awe (some starting in the '90s)

Zhang Ke Jia
Michel Gondry
Alfonso Cuaron
Apichatpong Weerasethakul
Nuri Bilge Ceylan
Iñárritu
+1 Steve McQueen
Edgar Wright
Ben Wheatley
+1 Kenneth Lonergan
Charlie Kaufman
Chan-wook Park
Shane Meadows (probably the closest to meeting the criteria)
Wes Anderson (ditto)
Masaaki Yuasa
Peter Strickland
Carlos Reygadas

probably some tv directors worthy of such company.

I just remember how satisfying Drive and Shame were. Good year.