Combine water, 1/2 cup flour, yeast, and sugar in a bowl; stir well and let sit until mixture is foamy and bubbling, about 20 minutes.
Stir 2 cups flour, 2 tablespoons olive oil, and salt into yeast mixture using a wooden spoon until dough holds together; turn onto a lightly floured work surface.
Coat the inside of a large bowl with 2 1/2 teaspoons olive oil.
Knead dough, adding small amounts of flour as needed, until soft and slightly sticky, about 10 minutes.
Form dough into a ball and place in the prepared bowl; drizzle about 1/2 teaspoon olive oil over dough and spread to coat entire ball. Cover bowl with a clean kitchen towel and place in a warm area of the kitchen until dough is doubled in size, about 2 hours.
Punch dough down and place in a large resealable plastic bag; refrigerate 8 hours to overnight. When ready to use, remove dough from refrigerator and bring to room temperature before using.