Cook spaghetti in a large pot of boiling salted water, for about 2 minutes less than the package instructions. You will finish cooking the pasta later in the recipe. Drain the pasta, reserving 1 cup pasta cooking liquid
Heat oil and butter in a large skillet over medium-high heat. Cook shallot and red pepper flakes, stirring often, until shallot is softened, about 2 minutes. Add tomatoes and cook, stirring often, until tomatoes are soft and juicy, 5–8 minutes.
Add lobster meat to skillet and toss to coat. Add pasta and ½ cup reserved pasta cooking liquid; season with salt and pepper. Cook, tossing constantly, until sauce thickens and coats pasta, about 2 minutes.