Preheat oven to 350ºF. In a medium bowl, beat cream cheese, sugar and vanilla extract with a wooden spoon until well blended. Chill the filling while you make the cookies.
Place chocolate in a microwave-safe bowl and microwave on 50% power for 2 minutes. Stir, then return to microwave at 50% power in 30 second intervals until chocolate is about 80% melted; remove and stir to finish melting it. Stir in sugar while the chocolate is still hot, and then the eggs, one at a time (stirring to fulling incorporate before adding the next). Stir in vanilla.
In a separate bowl, whisk together the flour, cocoa, baking powder and salt. Add to chocolate mixture and stir well to fully incorporate.
Roll cookie dough by tablespoonfuls into balls between your palms, then drop onto a parchment-lined baking sheet 2" apart. Make an indent in each cookie using your index finger, then drop cream cheese by about a teaspoonful onto each. Bake for about 8 minutes, or until edges are cracking but centres are still soft.