In a small bowl, combine active dry yeast, sugar, and ¼ cup warm water. Stir together until sugar and yeast have dissolved. Let sit until foamy and bubbly, about 5-6 minutes.
In a large bowl, whisk together flour, salt, yeast mixture and 1 ¼ cup warm water until a shaggy dough forms. Cover the bowl with a dish towel and set in a warm place and allow to rest until doubled in size, about 3 hours.
While dough rests, roast the garlic. Preheat the oven to 400F. Rub off excess layers of papery skin from each garlic head, and slice off top 1 inch to expose the cloves. Place each head on a small square of foil, then divide and drizzle 1 tablespoon of oil over top of each garlic head. Wrap the garlic tightly in the foil and bake until fragrant and tender, about 45 minutes. Remove from oven and allow to cool completely.
Once cool, carefully squeeze the bottom of each garlic head to remove each clove from their individual pockets. Place in a small bowl and set aside.
Flour a clean surface, turn out dough, and gently punch it down. Push each clove into the bread, making sure to space them out. Fold over the dough until the garlic has fully incorporated and the dough is smooth, about 4-6 times. Sprinkle on extra flour as needed. Place the dough onto a sheet of parchment, place a clean bowl, cover with a dish cloth, and allow to rise again for about 1 hour, or until doubled in size again.
Place a large Dutch oven in the oven (with racks arranged in the middle, then preheat to 450F. Once preheated, remove the pot from the oven and using the parchment paper as a sling, carefully transfer dough into the pot. Quickly drizzle remaining tablespoon of olive oil, sprinkle with flaky sea salt, then replace the lid.
Place Dutch oven back into oven and bake for 30 minutes, without removing the lid. Uncover and bake for an additional 10-12 minutes, until the top is crusty and golden brown.
Remove from the oven and transfer bread to a cooling rack. Wait for at least 20 minutes before slicing and serving.