Preheat to 400°F (190°C). Line a baking sheet with parchment paper. Gently mist or brush parchment paper with water to dampen the surface to help choux puff up.
In a medium saucepan, combine butter, water, milk, salt, sugar, and vanilla. Heat until butter melts and mixture simmers. Be careful not to boil at this stage, or your mixture may curdle.
Add the sourdough starter all at once. Stir vigorously WITH A WHISK over medium/low heat for 4-6 minutes, until a film forms on the bottom of the pan and dough pulls together. Your consistency should look like mashed potatoes.
Transfer dough to a bowl or stand mixer. Let cool for 7–10 minutes, until warm but not hot.
While dough is cooling, beat your eggs in a bowl lightly. When dough has cooled, add about 50 g (around 1/4th of the eggs) into the dough and beat into dough with stand mixer or hand beaters. It's important to add the eggs gradually to avoid adding too much egg. Continue adding eggs into dough and beating around 30 seconds with each addition. I like to stick with adding 1/4 of the eggs at a time. You will use more or less egg depending on how much your dough has been dried out during the cooking phase.