Bring a large pot of water to a boil over high heat; season generously with salt. Cook lobster, excluding the tail pieces, for 5 minutes; using a slotted spoon, remove from water. Reserve 1/4 cup cooking water. Set aside.
Bring another large pot of salted water to a boil. Add spaghetti, and cook, stirring occasionally, according to package directions. Drain spaghetti, reserving 1 cup cooking water; set aside.
In a large saute pan, heat 4 tablespoons oil over medium-high heat. Add lobster tails and cook for 4 minutes; reduce heat to medium-low. Add garlic and chiles; cook for 3 minutes. Add reserved lobster. Add tomato sauce; cook until heated through. Add pasta and remaining 2 tablespoons oil along with reserved cooking water, if necessary; toss to combine. Serve garnished with mint.