Mix 2 tablespoons sugar with water until dissolved. Add yeast and let sit until foamy, 5 to 10 minutes.
Meanwhile, mix remaining sugar, flour, and salt in a large mixing bowl. Combine sourdough starter and egg yolk in a separate bowl. Whisk egg white lightly and reserve in the refrigerator.
Pour yeast and sourdough mixtures over the flour mixture. Mix with a wooden spoon until dough starts to form and all dry ingredients are incorporated. Turn out onto a work surface and knead into a smooth ball.
Spray the inside of the bowl with cooking spray. Place dough in the bowl and cover with a cloth. Let rise for 1 hour.
Punch down dough and divide into 12 portions. Roll out into small balls; shape into 3- to 4-inch-long sticks. Divide bread sticks between 2 greased or nonstick pans. Cover with cloths and let rest for 45 minutes.
Preheat the oven to 350 degrees F (175 degrees C).
Bake in the preheated oven until light brown, 15 to 20 minutes. Remove 1 pan at a time and brush tops with egg white. Continue baking until medium brown, 5 to 10 minutes more.