Combine the bread flour, sugar, yeast and kosher salt in the bowl of a stand mixer and combine. While the mixer is running, add the water and 2 tablespoons of the oil and beat until the dough forms into a ball. If the dough is sticky, add additional flour, 1 tablespoon at a time, until the dough comes together in a solid ball. If the dough is too dry, add additional water, 1 tablespoon at a time. Scrape the dough onto a lightly floured surface and gently knead into a smooth, firm ball.
Grease a large bowl with the remaining 2 teaspoons olive oil, add the dough, cover the bowl with plastic wrap and put it in a warm area to let it double in size, about 1 hour. Turn the dough out onto a lightly floured surface and divide it into 2 equal pieces. Cover each with a clean kitchen towel or plastic wrap and let them rest for 10 minutes.