Make the dough: Combine the yeast, sugar and 1 cup warm water (100 degrees F to 105 degrees F) in a food processor and pulse once. Let sit until foamy, 5 to 10 minutes. Add the olive oil and pulse to combine. Combine the flour and salt in a medium bowl, then add to the food processor and pulse until the dough comes together. Transfer to a lightly floured surface and form into a ball. Place in a lightly oiled bowl, cover with plastic wrap and let rise in a warm place until doubled in size, 2 to 4 hours.
Make the rolls: Stretch the dough into an 8-by-12-inch rectangle on a lightly floured surface. Spread the marinara sauce on the dough all the way to the edge; top with the pepperoni and 1 3/4 cups mozzarella. Starting from a long side, tightly roll up the dough, slightly stretching as you go, to form a 14-inch-long log.
Coat the bottom and sides of a 9-inch square baking dish (preferably glass or light metal) with cooking spray, then line the bottom and sides with parchment paper, leaving a 1-inch overhang. Cut the log crosswise into 9 rounds. Arrange cut-side up in the prepared pan in 3 rows of 3, slightly touching. Cover with foil and set aside, 20 to 30 minutes.
Preheat the oven to 375 degrees F. Bake the rolls, covered, until the cheese begins to melt and the rolls are puffy, about 20 minutes. Uncover and continue baking until golden brown, about 20 more minutes. Sprinkle with the remaining 1/4 cup mozzarella and the parmesan; continue baking until the cheese melts, about 7 more minutes. Let cool slightly in the pan. Top with basil.