Make a poolish: Put 1 1⁄2 cups of the flour, sugar, yeast, and 1 cup warm water into a large bowl and stir well with a wooden spoon to combine. Let the mixture sit in a warm spot until bubbly, about 30 minutes.
Add remaining 2 1⁄4 cups flour, 3⁄4 cup of the oil, and salt to the poolish and stir until a dough forms. Turn the dough out onto a lightly floured surface and knead until smooth and elastic, 5–7 minutes. Grease a large clean bowl with the remaining 1 tbsp. oil, place dough in the bottom, and cover with a clean towel. Set the dough aside in a warm spot to let rise until doubled in bulk, 3–4 hours.
Preheat oven to 400°. Gently turn dough out onto a large sheet of parchment paper and gently stretch it with your fingers to form a 12" circle. Using a small, sharp knife, cut out five 2"-long slits, each about 1" wide, starting from the center of the bread and cutting toward the edge, so that the dough will resemble a sand dollar (discard dough scraps or bake them separately as a cook's-bonus nibble). Using your fingers, gently stretch the holes open a little wider so that they won't close up completely when bread is baked. Carefully transfer the dough—on the parchment paper—to a large baking sheet and bake until golden brown and puffed, about 15 minutes. Remove the bread from the oven and immediately brush the top and sides with 2 tbsp. hot water to soften the crust. Transfer the bread to a rack to let cool, or serve warm, if you like. (Remember to tear the bread into pieces—tradition says that cutting it might sever your friendship with the people at the table.)