For the dough: Fill a small bowl with hot tap water. Let the water sit in the bowl for 2 minutes to warm the bowl, then dump it out. In the warmed bowl, combine the yeast, sugar, 1 tablespoon of the flour and 1/4 cup warm water (105 to 115 degrees F). Let stand about 5 minutes. (The mixture will be creamy and slightly bubbly; if it isn't, start over with new yeast.)
In a large bowl, mix together the kosher salt and remaining 2 1/2 cups flour. Make a well in the center and stir in the yeast mixture, olive oil and 3/4 cup warm water (105 to 115 degrees F) with a wooden spoon. Mix until the dough starts to come away from the sides of the bowl (it will be sticky). If the dough looks dry, add an additional 2 tablespoons warm water. Turn the dough out onto a lightly floured surface; knead with floured hands until soft and elastic, about 8 minutes. Form into a ball. (If using a stand mixer, mix on medium speed until soft and elastic, 4 to 7 minutes.) Brush another large bowl with olive oil. Add the dough and roll to coat with the oil. Tightly cover with plastic wrap and let rise in a warm place until doubled in size, 1 to 1 1/2 hours.
Stretch the dough into the prepared baking dish. Cover with plastic wrap and let rise 30 minutes.
Preheat the oven to 425 degrees F. Spray a 9-by-13-inch baking dish with nonstick cooking spray.
For the topping: Use your fingertips to dimple the dough. Brush with the oil, top with the sun-dried tomatoes and rosemary and sprinkle with flaky sea salt.
Bake until golden brown, about 25 minutes. Remove from the baking dish and allow to cool slightly before slicing into squares. Finish with a drizzle of olive oil and more flaky sea salt.