In a large bowl, stir together the flour, salt, yeast and sugar.
Pour in the warm water and mix with a rubber spatula, until a sticky dough is formed.
Add 2 tablespoons of olive oil to the bowl and coat the outside of the entire dough ball, flipping it over inside the bowl.
Cover the bowl with a clean damp tea towel (or plastic wrap) and place in the refrigerator for at least 12 hours (overnight) or up to 3 days. To make the quick version, see Notes.
Grease a 9x13 inch pan with vegan butter. Pour 2 tablespoons of olive oil in the center of the pan.
Transfer the dough to the prepared pan, and roll the dough ball in the oil to coat it all over. Cover the pan and let the dough rise for 2-3 hours at room temperature, until the dough covers most of the pan and has expanded.
Preheat the oven to 425 degrees F.
If using fresh rosemary, sprinkle it over the dough. Drizzle the last 2 tablespoons of olive oil on the dough. Rub your hands lightly in the oil, then use all of your fingers to press straight down and create deep dimples, all the way down. Gently stretch the dough to fill the pan, if needed as you make the holes. Sprinkle with flaky sea salt.
Bake in the center rack of the oven for 25-30 minutes, until golden.
Carefully transfer the focaccia to a cooling rack. Let cool for 10 minutes, or longer, before slicing and serving.