In a bowl (or the bowl of a stand mixer) combine 1 cup warm water, 2 teaspoons granulated sugar, and 1 package active dry yeast. Stir, then let sit 10 minutes, until the mixture is foamy.
Add 2 to 2 1/2 cups all-purpose flour (start with 2 cups and add up to 1/2 cup more if needed), 1/2 teaspoon salt, and 1 tablespoon olive oil to the yeast mixture. Mix with the mixer and dough hook or knead by hand for about 5–7 minutes, until the dough is smooth and elastic.
Lightly cover the bowl and let the dough rise in a warm place until doubled in volume, about 1 hour.
Turn the risen dough out onto a lightly floured countertop. Roll it into a large rectangle no more than 1/2 inch thick.
In a small bowl stir together 6 tablespoons melted unsalted butter, 2–3 teaspoons garlic powder, 1 teaspoon dried parsley flakes, and a drizzle of extra virgin olive oil. Reserve half of this butter mixture for brushing the tops later.
Use a pastry brush to spread about half of the butter mixture evenly over the rolled dough. Sprinkle 1/4 cup of the freshly grated Parmesan cheese over the buttered dough.
Roll the dough tightly lengthwise into a log (like a cinnamon roll log). Use a sharp serrated knife to cut the log into pieces about 1 1/2 inches thick.
Place the rolls cut-side up on a greased baking sheet, leaving a little space between them. Cover the rolls with plastic wrap that has been sprayed with cooking spray (so it won't stick) and allow them to rest for about 30 minutes.
While the rolls are resting, preheat the oven to 400°F (205°C).
Bake the rolls at 400°F for a total of 12–14 minutes. At about the 9-minute mark, remove the rolls from the oven, brush the tops with the reserved butter/garlic mixture, sprinkle the remaining 1/4 cup Parmesan cheese over them, and return to the oven to finish baking until golden and cooked through.
Remove the rolls from the oven, let cool a few minutes, then serve.