NOTE: the sourdough starter can only be used if the butter is browned. Browning the butter removes the water from it which is then replaced by the water in the sourdough starter. If you don't brown the butter and still use the sourdough starter the cookies will end up a very different texture!
You can also skip browning the butter and the sourdough starter altogether. The cookies will still be very tasty.
In a medium saucepan, cook the butter over medium high heat until the butter melts, bubbles and then foams. Watch it closely and when the milk solids start to brown, remove from the heat and pour the browned butter into the bowl of a stand mixer or large mixing bowl. Allow the butter to cool completely. You should have approximately 94g of butter left (depending on the fat content of the butter you used).
Add the granulated and golden brown sugars to the bowl with the butter and using the paddle attachment beat on high for 3 minutes until light in colour and fluffy. Add the baking soda and salt then beat again briefly until combined. Add the egg and vanilla extract. Beat again until combined. If you are using sourdough starter, add it and mix until the dough is smooth and fully combined.
Add both the all purpose and rye flour and mix on low until the dough starts to form together. Add the chocolate chips and mix on low until no dry patches of flour remain. Wrap the dough or place it into a food storage container and chill it in the fridge or a minimum of 2 hours up to overnight.
Split the dough into 16 equal pieces (approximately 44g each or just eyeball it). Using your hands roll each piece into a ball then flatten into a disc about 1" thick. Place all the discs on a parchment lined baking sheet and place in the freezer until frozen solid.
Preheat the oven to 350 degrees F and adjust the baking racks so there are two near the middle of the oven. Line two baking sheets with parchment paper or baking mats and equally space 8 discs of dough on each baking sheet. Sprinkle with course or flaked salt if you like. Bake both baking sheets at the same time for 8 minutes, then switch the baking sheets on the racks and rotate each one around. Bake for another 7 minutes or until the edges are golden and the middle is just set.
Remove the baking sheets from the oven and lightly bang them on the stove top or counter to help the cookies collapse. Allow the cookies to cool on the baking sheets for approximately 10 minutes before moving them to cooling racks to cool completely.