Add sourdough starter, water, honey, softened butter, and 1 1/2 cups flour to the bowl of a stand mixer fitted with a dough hook. Mix until a smooth dough forms.
For the last 1/2 cup flour, add a little bit at a time until the dough begins to pull away from the bottom and sides of the mixer. Only use as much as necessary to form a workable dough.
Let your dough sit for 15 minutes.
Add your sea salt then continue kneading on low speed for 5 minutes or until the dough is smooth and elastic.
Transfer the dough to a large greased bowl and cover with a tea towel. Let the dough ferment for 8-12 hours. You'll know it's ready when it's doubled in size. *to make this an overnight recipe, you can allow the dough to ferment for 4-6 hours, put it in the fridge until you're ready to bake. When you're ready to bake, bring it out several hours before shaping to allow it to proof at room temperature.
Shape your dough by rolling it out on a floured surface (make sure your surface can handle a pizza cutter). Use a rolling pin to shape it into a 12x7 inch rectangle.
Using a pizza cutter, slice the breadsticks into 10-12 breadsticks.
Transfer breadsticks to a parchment-lined baking sheet, leaving about 1-inch space between each one. I split mine onto 2 baking sheets.
Preheat your oven to 400F and allow breadsticks to proof in a warm place for a couple of hours until they puff up. I like to put the pan on top of the stove while the oven preheats since it gets quite warm there.
Bake your breadsticks on the center rack for about 10 minutes. They should just barely start to brown on top.
Remove from the oven and immediately brush with melted butter and sprinkle with the garlic-parmesan mixture.