Roast the head of garlic: Slice off the top so the cloves are exposed inside. Place, clove side up, in a baking pan, pie dish or on a baking sheet and drizzle with a bit of olive oil. Top it with some fresh ground pepper and sea salt. Cover with aluminum foil and roast at 375°F degrees for about 35-40 minutes. The cloves will be very soft inside. Set aside.
Make the dough: In a large mixing bowl or the bowl of your stand mixer fitted with a hook attachment, combine the yeast and warm water. Stir it around and let sit for 5 minutes of until the yeast is foamy and dissolved. Add flour, olive oil, salt, and sugar. Mix by hand with a wooden spoon or with the mixer on low speed for 1 minute.
Once all of the ingredients are mixed, knead for 7 minutes by hand or with your mixer on low-medium speed. If your dough is too wet, slowly add up to 1/4 cup more flour. After kneading, your dough should be smooth and elastic. Poke it with your finger - if it slowly bounces back, your dough is ready to rise. If not, keep kneading.
Shape the dough into a ball and place in a large bowl that has been coated lightly with olive oil. Turn it over to coat all sides. Cover tightly with plastic wrap and allow to rise in a warm environment to rise, about 1 1/2 hours or until doubled in size.
Punch the dough down to release the air. Divide the dough into two. Roll each half into a ball and let rest in two separate bowls, lightly covered, for 10 minutes.
Preheat oven to 375°F degrees. Line two large baking sheets with parchment paper or silicone baking mats. Set aside. Chop up the roasted garlic and combine it with the melted butter in a small bowl. Add salt and pepper to taste. Set aside.
Assemble: Roll each ball of dough out into a 7x10-inch rectangle on a lightly floured surface or on a silicone baking mat. Top each dough rectangle with melted butter/roasted garlic mixture. Then top with grated Parmesan. Using a very sharp knife or pizza cutter, cut rectangles into long strips, about 1 1/2 inches in width. Then, twist each strip up and top with a little more Parmesan cheese if desired. Twisting is optional.
Bake breadsticks in two batches - 7 breadsticks per baking sheet. Bake for 16-17 minutes. Then, remove from the oven, push the 7 breadsticks together so they're all touching, and top with 1/2 cup of shredded cheese. Return to the oven for about 4-5 more minutes or until they've reached your desired doneness. Like them extra brown and toasty? Keep in for a few minutes longer. Repeat with second batch and remaining 1/2 cup of cheese. (If you don't want cheese on top of them, then just bake the breadsticks for 20-25 minutes or until they've reached your desired doneness.)
Remove from the oven and enjoy! Cover leftover breadsticks tightly and keep at room temperature for up to 2 days. Baked breadsticks freeze well - up to 2 months.