Preheat the oven to 425 degrees F. Line a rimmed baking sheet with parchment paper.
Heat 1 tablespoon of the olive oil in a medium saucepan over medium heat. Add the onions and 1/2 teaspoon of the oregano, season with 1 teaspoon salt and cook, stirring occasionally, until soft, about 4 minutes. Add the garlic and cook until fragrant, about 1 minute. Add the tomato puree, 1/4 cup water, 5 torn basil leaves and the red pepper flakes. Bring to a simmer and cook until slightly thickened, about 4 minutes. Season with salt and pepper.
Meanwhile, heat the remaining tablespoon olive oil in a large skillet over medium-high heat. Add the sausage rounds and brown on both sides, about 4 minutes. Transfer to a paper-towel-lined plate. Mix the Parmesan with the remaining 1/2 teaspoon oregano in a small bowl.
Divide the pizza dough into 8 portions. Flatten into 4-inch squares either by stretching with your hands or using a rolling pin. Spoon 1 tablespoon of sauce onto a square and spread around. Top with 1/4 cup of the Italian cheese blend, one-eighth of the sausage and 1 basil leaf. Wrap the dough around the filling and pinch the edge and ends to seal. Place seam-side down the prepared baking sheet. Repeat with the remaining dough and ingredients.
Brush the rolls with olive oil and sprinkle with the Parmesan oregano mixture. Cut 2 slashes across the top. Bake until golden and bubbly, 20 to 25 minutes. Serve warm with the remaining sauce on the side for dipping.