Peel the shrimp and save the shells. Put the shrimp in a bowl with the chopped rosemary, 3 cloves of the chopped garlic, and 3 tablespoons of the olive oil. Toss well, cover, and marinate in the refrigerator for about 2 hours.

While the shrimp is marinating, make the shrimp stock. Heat 2 tablespoons of the olive oil in a large stockpot of medium heat. Add the shrimp shells and sauté, stirring occasionally, for about 5 minutes. Add the carrots, onion, 3 more cloves of chopped garlic, and 2 more tablespoons of olive oil; cook, stirring occasionally, for 7 to 8 minutes longer. Add the broth and wine, bring to a boil over high heat, and then reduce the heat and simmer for 15 minutes. Pour through a fine-meshed sieve over a large bowl to strain out the solids; then, return the stock to the stockpot and warm over medium heat.

For the risotto, in another stockpot or large saucepan, sauté 3 more cloves of chopped garlic in 2 tablespoons olive oil over medium heat for about 1 minute. Pour off the marinade liquid from  the shrimp and cook 2 to 3 minutes longer. Add the rice, stirring to coat it, and cook for 1 minute. Add the vermouth and begin stirring the rice mixture continuously with a wooden spoon. As the rice absorbs the vermouth, begin adding the shrimp stock a ladleful at a time, stirring continuously after each addition until the liquid is absorbed before adding more stock. Continue this process until the rice is al dente, tender but still slightly chewy, and surrounded by a thick, creamy sauce, about 25 minutes. Turn off the heat. Stir in 2 tablespoons butter, 1 cup grated cheese, cheese and a splash of white wine.

In a sauté pan over high heat or under a hot broiler, cook the shrimp until just pink and firm, 1½ to 2 minutes per side. Spoon the risotto onto individual heated serving plates or shallow pasta bowls and arrange 4 or 5 shrimp on top of each serving. Garnish each serving with more grated cheese and a rosemary sprig.