Bring a large lobster pot filled with salted water to a roiling boil.
Plunge the lobsters in, head first and cook for about 10 minutes until lobsters turn very pink.
Remove the lobsters from the water and cool completely. Do not throw out the lobster water.
Once the lobsters have cooled, remove the meat and set aside.
Bring one quart of the lobster water to a boil and cook the fresh pasta for about 3 minutes until al dente. Remove from the water and set aside the pasta.
In a heavy bottomed skillet heat 1/4 cup olive oil and 5 Tbsp. butter. Add 1/2 tsp. tarragon and all of the lobster meat. Stir well. Cook for about 5 minutes until the lobster is thoroughly coated with oil and butter.
Add 6 Tbsp. lobster water to even out the sauce and then stir in the cooked pasta and cook for another 10 minutes stirring until well combined.
Serve in big bowls with steamed asparagus or veg of choice.