Make the sponge by stirring together the water and sugar in a medium-sized bowl. Add the yeast and stir again to dissolve. Set aside to proof for 5 minutes. There should be bubbles and foam on top after 5 minutes. If there is no bubbles, the yeast is dead and you need to start over with fresh live yeast.
Once the yeast has proofed, add 1 ½ cups of the flour (210 g) and stir to form a thick paste. Cover with plastic wrap and place in the refrigerator overnight. The next day the sponge should have grown and will look "stringy" in texture.
After the sponge has chilled overnight, scoop it out and place it in the bowl of a stand mixer fitted with a dough hook and add the remaining 1 ¼ cups (175 g) of flour and the salt to the bowl. Stir the dough with the hook on low, stopping occasionally to scrape the sides with a spatula to incorporate the dry flour.Once the flour has been absorbed increase the speed to medium low and knead the dough for 3 to 4 minutes. The final dough should be smooth and slightly tacky but not sticky. If it is sticky, add 1 to 2 tablespoons of flour in the kneading process.
If you don't have a mixer or want to knead the dough by hand, just add the flour and salt directly to the bowl with the sponge and mix with a wooden spoon until the flour is mostly absorbed. Then scrape the dough onto a flour dusted surface and knead the dough by hand for about 4 to 5 minutes, or until the dough is smooth and slightly tacky. If the dough is too sticky, add 1 to 2 tablespoons of flour during the kneading process but try not to add too much as that will dry out the bread.
Once the dough is kneaded and smooth, grease a 9-inch cast iron skillet with 1 tablespoon of the sunflower oil. Divide the dough into 8 even pieces (about 75 grams each if you have a scale) and form round balls with each of them.
Place 7 rounds of dough in a circle, around the perimeter of the pan, and one dough in the middle. Brush the top of the dough with the remaining sunflower oil. Brush any residue oil onto a piece of plastic wrap and cover the pan with the plastic, oil side down. The oil will prevent the dough from sticking to the plastic wrap.
Place the pan in a warm area and let it rise until the balls have doubled in size, and are touch each other, about 45 to 75 minutes depending on how warm or cold the area is where you let the dough rise.About 15 minutes before the dough is ready, preheat the oven to a very hot 450°F. Once the dough is ready, remove the plastic wrap. Beat the egg yolk and the water together and gently brush the egg wash over the top of the balls of dough. Bake in the oven for 14 to 16 minutes or until the dough is golden brown on top. If you have an instant read thermometer, the inside should register 190 to 200°F.
While the bread is baking make the garlic oil. Cut off the root of the green garlic and discard it. Finely chop the green garlic and the flat-leaf parsley. Place in a bowl and pour the sunflower oil over it, mixing it together. Once the bread is done, take it out of the oven and immediately brush the top of the bread with the garlic oil, making sure you get as much of the solid green garlic and parsley bits as possible onto the bread. Sprinkle with a couple pinches of kosher salt and serve immediately.
