Search found 3 matches: Charlie Kaufman

The only director that goes by this criteria for me is Charlie Kaufman with Synecdoche NY (t10) and Anomalisa (t9). Two directors that were very close (with the t8 films on the cusp of t9) were Greg McLean with Wolf Creek (t10) and Wolf Creek 2 (t8) and Ben Wheatley with Kill List (t10) and A Field in England (t8).

I'm sure (hopefully) in the future Alejandro González Iñárritu would make my list (if he ever makes a dramedy as good as t10 Birdman instead of another mopey sad-fest) along with S. Craig Zahler (who made Bone Tomahawk and hopefully his next film is also in my t10).

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.