While it's not impossible to find an original show on Broadway, the majority tend to be Disney Shows, "Jukebox Musicals" (musicals based on the music of a band or musician), and classics featuring current TV or Movie Stars. The rising costs of big named actors, costumes, and theatre rentals make it impossible to mount a show that isn't guaranteed to bring in a large audience.
Playwrights Horizons (just beyond the bright lights on 42nd St., past 9th Ave.) features contemporary American playwright's new work. MCC Theater also supports contemporary playwrights, specifically work that explores "emotional and moral issues of contemporary life" (just actors on stage reading from the script) and producing them as they develop (their last season's 'Hand To God' opened on Broadway just last month) and the Manhattan Theater Club states in their mission that they're "constantly seeking new ways to innovate [and keep] theater alive and relevant". All three of these theaters (non of which mentioned on Broadway.com) offer talk-backs after specific shows where the audience can dissect the play along with playwrights, directors and other creatives. They also host readings and workshops for new works at a very elementary stage.