Jazz Haus is weekly on Monday. Not every Monday has Burlesque, but it looks like about 3 out of 4 shows do, albeit typically only one performer.
From their website:
Join us as we celebrate the vibrant, hedonistic Weimar era with our ‘neue’ weekly ‘Tanz Kabarett’ at Bierhaus NYC featuring gotham’s best Hot Jazz n’ Swing bands!! Plus swing DJ’s, classic burlesque, free dance lessons… und exzellent German food n’ drink specials – it’s like ‘Radegast’ but in midtown, just 2 blocks from Grand Central… and NO COVER!!