Bloom the yeast: In a large bowl, stir 1 cup warm water (105–110°F) with 1 tsp sugar to dissolve. Sprinkle in the active dry yeast and let stand until foamy, 5–10 minutes.
Make the dough: Add 3 cups of the all-purpose flour, 1.5 tsp kosher salt, and 2 tbsp olive oil to the bowl. Mix until a shaggy dough forms, then knead on a lightly floured surface 8–10 minutes until smooth and elastic, adding only enough of the remaining flour to prevent sticking.
First rise: Lightly oil a clean bowl using a little of the remaining olive oil. Place the dough in the bowl, turn to coat, cover, and let rise in a warm spot until doubled, 60–75 minutes.
Make the sauce: In a small saucepan, combine the tomato paste, 0.5 cup water, 1 tbsp olive oil, remaining 1 tsp sugar, 0.75 tsp kosher salt, dried oregano, dried basil, garlic powder, and red pepper flakes. Simmer over low heat, stirring, until thick and spreadable, 3–5 minutes. Cool to room temperature.
Preheat the oven to 425°F (220°C). Line a rimmed baking sheet with parchment.
Roll and fill: Punch down the dough and roll on a lightly floured surface (use about 0.25 cup of the reserved flour) to a 12×18-inch rectangle. Spread 0.5 cup of the cooled sauce in a thin, even layer, leaving a 0.5-inch border. Sprinkle the mozzarella evenly over the sauce, then distribute the pepperoni.
Roll and slice: Starting from a long edge, roll the dough up tightly into a log, pinching the seam to seal. Trim the ends if needed, then cut the log into 12 equal pieces with a serrated knife.
Arrange and bake: Set the pieces cut side up on the prepared sheet, spaced about 1 inch apart. Brush the exposed dough lightly with the remaining olive oil. Bake until golden brown and the cheese is bubbling in the centers, 15–20 minutes, rotating the pan halfway through. Rolls are done when the bottoms are lightly browned and the centers read molten and bubbly.