Peel one large head of garlic; cut large cloves into marble-sized chunks. Place in a small saucepan and cover with olive oil. Simmer ~15 minutes until garlic is tender. Remove the garlic from the oil, save the oil.
In a stand mixer combine
3 1/2 cups unbleached all purpose flour
1 1/2 tsp yeast
1 1/4 tsp salt
With the motor running add the 1 cup potato water (ideally still warm, but not so hot it will kill the yeast) and mix until the dough comes together in a shaggy ball. Add the potato and mix just until combined, then 2 tbsp of the garlic-infused oil. Mix until it comes together.
Transfer the dough to an oiled (use some of the reserved garlic oil) rising container, cover, and allow to rise 35-45 minutes until doubled in size.
Tear the dough into two halves and form into rounds. Place the rounds on an oiled cookie sheet and drizzle a little oil on top. Cover with saran wrap and allow to rise again, 35-45 minutes.
Preheat the oven to 425 (you can use the top of the oven as a warm place to rise the dough) at least 30 minutes ahead of time.
Once the second rise is complete, poke holes in the top of the dough with a wet finger, then tuck a chunk of garlic into each hole. Drizzle remaining olive oil over the top of the dough, allowing some to puddle in the holes, and then sprinkle with ~2 tbsp fresh rosemary leaves and ~1 tbsp coarse sea salt.
Bake for 23-25 minutes until golden brown.