Heat the chicken broth in a medium pot over medium heat until it starts to gently boil. Add rosemary sprigs, let them infuse for 5 minutes, then remove and discard them.
Gradually whisk the yellow pre-cooked cornmeal into the boiling broth, stirring constantly to prevent lumps.
Lower the heat to a simmer and cook the polenta, stirring occasionally, for 10–15 minutes until it thickens and easily pulls away from the pot's sides.
Season the polenta with salt, black pepper, and olive oil. Taste and adjust the seasoning if needed.
Pour the warm polenta into a greased baking dish and spread it evenly.
Sprinkle shredded mozzarella cheese on top of the polenta.
Broil the dish for 5–7 minutes, or until the cheese is melted, bubbly, and golden brown.
Serve the hot polenta as a versatile dish, paired with vegetables, grilled meats, or roasted chicken.