Combine 1 cup warm water (110 degrees F), the sugar and yeast in a large bowl. Let stand until foamy, about 5 minutes. Add 1 tablespoon olive oil, the flour and fine salt; stir to form a sticky dough. Turn out onto a floured surface and knead until smooth, about 10 minutes, dusting with flour.
Brush a large bowl with olive oil; add the dough, loosely cover with plastic wrap and set aside until doubled in size, about 1 hour, 30 minutes.
Brush a rimmed baking sheet with olive oil. Stretch the dough on the baking sheet. Cover with a towel; set aside until doubled, 30 minutes.
Preheat the oven to 475 degrees F. Scatter the cheese, bell pepper and scallion whites over the dough. Bake until the crust is golden, 15 to 18 minutes. Toss the spinach and scallion greens with the remaining 1/2 tablespoon olive oil and the sea salt. Sprinkle over the flatbread; bake 2 more minutes.