Heat oil in a large wide pot or deep skillet over medium; sauté garlic 1-2 minutes until fragrant but not browned.
Add tomatoes and oregano; stir 2-3 minutes until tomatoes start softening.
Add broth; boil, submerge pasta, and stir gently.
Cook 8-10 minutes, stirring occasionally, until pasta is al dente and most liquid absorbed; pasta continues absorbing liquid as it sits.
Off heat, immediately stir in mozzarella and basil; season with salt and pepper. Cheese starts melting into creamy pockets.
Drizzle with balsamic glaze; top with extra fresh basil and optional Parmesan and serve warm.