To begin, make sure your starter is in the active, bubbly phase. Depending on the strength of your starter and the time of year (temperature makes a big difference), you most likely will want to feed your starter 6-12 hours before you start this recipe.
In a large bowl, add flour, sugar, and salt. Stir to combine.
In another bowl, combine sourdough starter, egg, and water. Mix until combined.
Gradually add flour to sourdough mixture. Knead by hand or with the dough hook of your stand mixer for 6-8 minutes, until smooth and elastic.
Cover with a damp towel and allow to sit at room temperature for about 8-10 hours. If you aren't baking right away, allow to rise at room temperature for 4 hours and then the dough can be placed in the fridge for up to 3 days.
Once the dough has finished the rising process, roll out to about 1 inch thick and cut into equal sized pieces and form into breadsticks. Allow to rise for 1-2 more hours.
Preheat oven to 400 degrees. Bake breadsticks for 10 minutes.
Melt butter and brush on to cooked breadsticks, then sprinkle garlic and salt on top. They are best served warm.