Melt the butter with the garlic over low heat. Add the lobster and gently butter baste until just cooked through, then remove and cut into chunks. Save the lobster-infused butter for your sauce.
Cook the fresh pasta for about 2–3 minutes, reserving some pasta water.
Add the pasta to the butter, then add in some lemon zest and a splash of pasta water to the pan. Remove from the heat and stir in the Pecorino and pasta water, adding more of either as needed until silky.
Fold in the lobster, finish with more Pecorino, lemon zest and chives, and that’s it.