Put a pot with about 2quarts water over high heat and bring to a boil. Lower the heat to maintain a very gentle simmer.
Peel and finely chop the onion. Put it in a heavy-bottomed braising pan with 2tablespoons of the butter. Place over medium-high heat and sauté until it turns a rich golden color, about 5 minutes.
While the onion is sautéing, finely chop enough of the rosemary leaves to measure 3teaspoons. When the onion is ready, add 1teaspoon of the rosemary and the bouillon cube to the onion and stir until the cube is completely dissolved, about 1 minute.
Add the rice and stir until it is well coated. Add about 1cup of the hot water and continue stirring. Add only enough water to produce the consistency of a rather thick soup and wait until all the liquid is absorbed before adding more. Season with salt and continue until the rice is al dente, 20 to 25 minutes. Add the remaining 2teaspoons rosemary and 1tablespoon butter and stir well.
Remove from the heat and stir in the Parmigiano. Serve at once.