Preheat oven to 300°F/150°C.
In an 8-inch square baking pan, cut an 8" x 12" piece of parchment paper to fit inside the pan, leaving enough to hang over two sides of the pan (to create a sling). Grease the two sides of the pan that are not covered by parchment paper.
Separate 2 of the eggs from their yolks (use egg whites for another purpose).
Melt butter.
If using vanilla bean, split lengthwise and scrape the seeds of the bean out. Add scrapings to butter.
Beat the 2 eggs and 2 yolks in a bowl with a mixer on medium speed until fluffy and light yellow, 2 to 3 minutes. Mix in sourdough starter.
In a separate bowl, combine sugars, cocoa powder and salt. Drop mixer speed to low and slowly introduce the dry mixture into the batter bowl, about 1/3 at a time.
Add butter and vanilla extract (if using) to bowl. Continue mixing batter. It will be ready when it's shiny with just small lumps.
Pour batter into the pan and bake 45 to 50 minutes.
Test doneness by inserting a toothpick in center of pan. When it comes out dry (no crumbs), it is ready. If has batter or crumbs, leave in a few more minutes. Remove from oven.
Let cool in the pan for 3 minutes, then lift brownies out using the parchment paper sling.
Cut into 16 pieces (with a pizza cutter or dough scraper) and let cool completely.