Pour flour onto a clean work surface and make a well in the middle. Mix eggs, egg yolk, and 1/2 teaspoon salt together in a bowl. Pour the egg mixture into the well; slowly incorporate flour into egg mixture until combined. Knead dough until smooth and elastic, 8 to 10 minutes, adding more flour as needed. Cover and let rest for at least 30 minutes.
Roll dough using a pasta machine or rolling pin; cut into noodles about 1/4-inch wide.
Bring a large pot of water to a boil; add 1 tablespoon salt. Prepare a large bowl of ice water. Add lobster tails to the boiling water. Cook for 3 to 4 minutes; drain and shock in the ice water. Slit the shells with kitchen shears and gently extract the meat.
Heat 1/4 cup water in a saucepan over medium heat to just under a boil. Add butter, piece by piece, using a kitchen thermometer to keep the sauce at 160 to 180 degrees F (71 to 82 degrees C). Add lobster tails; cook until meat is opaque yet soft and almost undercooked, 5 to 6 minutes. An instant-read thermometer inserted into the center should read at least 145 degrees F (63 degrees C). Reduce heat to low.
Bring a large pot of lightly salted water to a boil. Cook noodles in the boiling water until tender yet firm to the bite, 1 to 3 minutes; drain.
Divide pasta and lobster equally among serving plates. Add tomatoes to the butter sauce; cook until barely softened, about 2 minutes. Split the sauce and tomatoes equally over the pasta and lobster. Season with salt and pepper; garnish with basil leaves.