Bring a large pot of salted water to a boil. Add the pasta and cook until al dente, 3 to 4 minutes. Remove 1/2 cup pasta water from the pot and set aside; drain the pasta in a colander. Transfer the pasta to a bowl and toss with 1 tablespoon of olive oil, cover and set aside.

Meanwhile, heat the remaining 1 tablespoon of olive oil and 2 tablespoons of butter in a large skillet over medium heat. When hot, add the shallots and cook, stirring, until soft and fragrant, about 1 minute. Add the garlic and continue to cook and stir for another minute. Add the white wine, saffron and tomato paste and cook for 5 minutes longer. Add the lobster stock and the reserved pasta water and  bring to a boil. Cook until the stock is reduced by half, 5 to 7 minutes. Add the heavy cream and cook until reduced by half, about 5 minutes longer. Add the tomatoes, salt and pepper and reduce the heat to low. Cook just until the tomatoes release their liquid, about 3 minutes. Stir in the chives and the butter and whisk to blend. Add the chopped lobster and cook until just heated through. Toss the pasta with the sauce and cook for 1 minute longer, until the pasta is heated through.

Divide the pasta between 4 bowls, garnish with the Parmesan cheese and serve immediately.