Prep ingredients by mincing garlic, slicing shallots, halving cherry tomatoes, stripping tarragon leaves, zesting lemon, and snipping chives.
Bring a large pot of salted water to a boil, cook the linguine pasta until al dente following package directions, reserving 1 cup of pasta water before draining.
Cut lobster tails into 1-inch chunks and set aside.
Heat olive oil in a large pan over medium heat; sauté shallots and garlic for 2-3 minutes until softened and fragrant.
Add cherry tomatoes and cook for 2 minutes until softened.
Pour in white wine and stir, scraping bottom of the pan; simmer 1-2 minutes to reduce and evaporate alcohol.
Reduce heat to medium-low, add lobster chunks, tarragon leaves, and heavy cream; simmer gently for 3-4 minutes until lobster is cooked and sauce thickens.
Remove from heat and stir in butter and lemon zest until sauce is silky; then add lemon juice and season with salt, pepper, and nutmeg.
Add cooked pasta to the sauce and gently toss to combine; adjust sauce consistency with reserved pasta water if needed.
Serve pasta garnished with fresh chives, red pepper flakes, and parmesan shavings.