To make the stock: Steam the lobster tails on a rack over simmering water in a large covered pot until they turn a bright red-orange, about 8 minutes.
Use tongs to transfer them to a cutting board, then pour the cooking liquid into a 4-cup heatproof measuring cup; if needed, add enough water to make 4 cups.
Use kitchen shears to cut through the underside of the shells and remove the tail meat.
Slice into 1-inch medallions, place in a dish, and cover; set aside.
Heat a large skillet or Dutch oven. When hot, add the olive oil, lobster shells, and all the vegetables.
Sauté until the vegetables soften, stirring frequently.
Add the tomato paste to the center of the pan and cook it until it darkens in color.
Carefully whisk in the reserved liquid, plus another 2 cups of water, the paprika, and the bay leaves.
Bring to a boil, then simmer for an hour. Strain the broth into a clean saucepan, pressing down on the vegetables to extract all the liquid.
You should have 4 cups; if not, add enough water to make that amount.
To make the risotto: Soften the saffron in the vermouth; set aside.
Bring the lobster stock to a low simmer in a medium saucepan.
Heat a heavy sauté pan or Dutch oven over medium heat.
When hot, add the olive oil and shallots, sautéing them until soft, but don’t let them brown.
Add the rice to the pan and stir to thoroughly coat with the oil.
Cook until the rice is slightly translucent, about 7 to 10 minutes, stirring often.
Add the vermouth-saffron mix to the pan and cook until the liquid has nearly evaporated.
Ladle 1/2 cup of the hot stock into the rice mixture and stir continuously.
When the liquid is almost completely absorbed, stir in another 1/2 cup of hot stock.
Repeat until the rice is creamy yet al dente, about 25 minutes (you may not need all the stock).
Stir in 1/2 cup of the Parmigiano-Reggiano and the lobster medallions.
If the risotto is too thick, stir in any remaining stock or water, 1/4 cup at a time.
Season to taste—the risotto might taste salty enough from the cheese but will benefit from a few twists of a pepper grinder.
Serve immediately, topping each portion with a drizzle of olive oil and some chives and passing the rest of the cheese on the side.