Preheat the oven to 375ºF (190ºC) and line two baking sheets with parchment paper.
Whisk the flour, baking soda, and salt in a bowl until well combined. Set aside.
Place the brown sugar, white sugar, and butter in a second mixing bowl.
Use a hand or stand mixer to mix everything together on medium to high speed until thick and creamy, about 1 to 2 minutes.
Crack an egg into the butter and sugar mixture. Continue to mix on medium-high speed until the egg is well mixed in. Repeat with the second egg.
Add the sourdough starter and vanilla to the bowl. Again, mix until well blended, smooth, and creamy.
Switch to a wooden spoon or rubber spatula. Add the flour mixture to the butter mixture.
Mix the cookie dough by hand until the flour is about halfway mixed.
Add the chocolate chips to the bowl, and continue to mix by hand until the flour is just mixed in.
Use a medium-cookie scoop to scoop 12 portions of dough onto each prepared baking sheet. The dough portions should be about 2 inches apart from each other and from the sides of the pan.
Place the pan in the preheated oven and bake the sourdough starter cookies for 9 to 11 minutes, or until the edges are light gold.
Remove the baking sheets from the oven to cool on a wire rack for 10 minutes.
Transfer the cookies from the baking sheet to the wire rack to cool completely.