Pull out cream cheese and allow to come to room temperature (about 30 minutes).
Preheat oven to 400° with a rack in the center. Butter an 8-inch springform pan. Line with two sheets of overlapping parchment paper. This cake is meant to have rough edges, so don't worry too much about smoothing out the sides.
In a medium bowl combine eggs, egg yolks, heavy cream, salt, vanilla, and sourdough discard. Whisk until smooth.
In a large bowl, or the bowl of a stand mixer fitted with paddle attachment, combine cream cheese, sugar, and matcha powder (if using). Beat together on medium speed until smooth, about 2 minutes.
Gradually add sourdough mixture to cream cheese mixture on medium speed until combined. Scrape down the side of the bowl. Mix again just until smooth and evenly combined.
Pour batter into prepared springform pan. Place springform pan on a sheet pan. Bake on the middle rack of oven until deeply browned on top, about 60-70 minutes. Cake will still be very wobbly in the middle. Allow to cool for 10 minutes then unmold and transfer to a wire rack (leaving it on the base of the springform pan is fine). Cool completely (at least 1 hour) before removing parchment. Serve cool or at room temperature.