Bring a large pot of salted water to a boil. Cook fresh pasta until al dente, 2-4 minutes, depending on thickness. Or cook dry pasta according to package directions for al dente. Reserve ½ cup of the cooking water before draining.

Meanwhile, cut open lobster tails to remove the meat. Remove vein and chop lobster meat into bite-sized pieces.

Melt butter in a large skillet over medium heat. Turn heat to medium-high. Add lobster meat and season with salt and pepper. Cook, stirring often, until lobster is cooked through, about 2 minutes. Add garlic and cook until fragrant, about 1 more minute.

Remove lobster to a plate. Add wine to the same pan and scrape up any browned bits from the bottom of the pan. Cook for 1 minute for the alcohol to cook off.

Turn heat down to medium. Add cream and milk and stir to combine. Bring to a simmer and simmer until sauce is slightly thickened, about 5 minutes. Do not let boil. Taste sauce and add salt and pepper as needed.

Remove from heat and stir in parmesan cheese. Mix until cheese is melted. Combine cooked lobster, cooked pasta, and sauce in the skillet or in the pasta cooking pot. If the sauce is too thick, pour in a little bit of the reserved pasta cooking water to thin it.

Divide pasta between serving plates. Garnish with chopped parsley and serve with lemon wedges.