To Make the Focaccia: In the bowl of a stand mixer, add warm water, sugar and yeast to a large bowl. Stir gently then let stand for 8-10 minutes until foamy.
Fit stand mixer with the dough hook. To the yeast mixture running on low speed, add bread flour, all purpose flour, 1/3 cup olive oil, 2 Tbsp chopped rosemary and 1 Tbsp sea salt.
Raise the speed to medium and let mixer knead for 5-7 minutes or until elastic and the dough has pulled away from the sides of the bowl. (If the dough is sticky, add additional all purpose flour 1 Tbsp at a time just until the dough comes together.)
Turn dough out onto a lightly floured surface and turn just until it's no longer tacky. Oil mixing bowl, then place dough back into the bowl, cover with a damp towel and let rise in a draft free place for 1 - 1/2 hour or until doubled.
Pour reserved 1/3 cup olive oil into a 15 x 10 inch jelly roll pan tilting to fully coat bottom and sides. Gently release air from dough then place dough onto pan pulling and stretching to fit the pan, flipping once.
Using your fingers, press holes all over the surface of the focaccia, pressing the dough until you feel the bottom of the pan without pushing all the way through.
Drizzle the top with more olive oil (eyeball it) allowing it to pool in the holes. Sprinkle with Parmesan cheese and sprinkle with reserved 2 Tbsp rosemary evenly over the top.
Cover and let rise in a warm draft free place for 45 minutes or until doubled.
To Bake: Preheat oven to 425°F. Uncover focaccia then bake for 25-35 minutes or until puffed and golden.
Let stand until warm then cut into squares or slice and serve.