Whisk together warm water, sugar, and yeast. Let sit for 5-10 minutes, until foamy.
In a large bowl, stir together 2 cups flour, parmesan, rosemary, and salt. Pour in the yeast mixture and stir until flour is evenly moistened.
Gradually mix in more flour until you have a cohesive, shaggy dough. Turn out onto a lightly floured surface. Knead for 6-8 minutes, adding just enough flour to prevent sticking.
Once you have a smooth and elastic dough, shape into a ball and place in a lightly greased bowl. Cover and let rise in a warm place until doubled, about 1 hour.
Gently deflate dough and shape into a tight ball. Place on a parchment lined baking sheet, cover, and let rise again for 30-45 minutes, until puffy and nearly doubled.
Halfway through rising time, heat oven to 450°F. Place rack so there is enough room for a Dutch oven with lid.
Once the dough has proofed, carefully transfer the dough on the parchment to the Dutch oven. (It's OK if some parchment hangs over the edge.)
Bake with lid on for 30 minutes, then remove the lid and continue baking another 10-15 minutes, until nicely browned.
Carefully transfer the bread to a rack to cool completely before slicing.
Notes
If using instant yeast, skip the first step and add the yeast and sugar directly to the dry ingredients.
Add flour slowly, you might not use the full 4 cups.