Special equipment: a pasta roller and cutter
For the pasta dough: Create a mound of flour on a clean, flat work surface. Using your fingers, make a well in the center of the mound. Pour the eggs and egg yolks into the well; add the tarragon and 1 tablespoon water. Using a fork, whisk the eggs while incorporating the flour from the inside of the mound, working outward until the flour and eggs are combined. With the palms of your hands, knead the dough. If the dough seems dry and shaggy, dampen your hands with water and knead until the dough is moist but not wet. If it feels wet, add flour, 1 teaspoon at a time. Knead the dough until smooth and elastic, about 10 minutes. Wrap the dough in plastic wrap and allow to rest at room temperature for 30 minutes.
For the vodka cream sauce: Meanwhile, heat the butter and oil in a medium saucepan over medium heat until the butter is melted. Add the shallot, sprinkle with salt, and cook until softened, 5 to 7 minutes. Add the garlic and crushed red pepper and cook until fragrant but not burned, 1 to 2 minutes. Remove the pan from the heat before carefully adding the vodka and tomatoes. Then put the pan back over medium-high heat and simmer until the alcohol has evaporated, 5 to 7 minutes. Stir in the tarragon, basil and lemon zest. Then lower the heat to medium-low and add the heavy cream; cook for 3 to 5 minutes, until the sauce thickens. Cover and keep warm over low heat.
To make the linguine, remove the dough from the plastic wrap. Cut the dough into equal quarters and roll each into a small, thick log. Use your hands or a rolling pin to flatten the logs of dough. Feed one piece of dough through a pasta roller set to the widest setting. Fold the dough into thirds and continue to rolling, adjusting the setting to achieve desired thickness. Cover with a thin towel and repeat with the remaining dough logs. Cut the sheets into the preferred noodle length.
Switch the pasta roller to a linguine cutter. Feed each sheet through the cutter. Flour the noodles and then separate them, or hang them to dry to prevent sticking. Repeat with all of the pasta sheets.
Bring a large pot of salted water to a boil over medium-high heat. Add the fresh pasta and cook until al dente, about 3 minutes. Remove the pasta and transfer to the pan with the sauce. Turn off the heat and gently stir just to coat the noodles.
For the lobster: Heat the oil in a heavy medium skillet over medium-high heat. Add the lobster pieces, sprinkle with salt and cook until the lobster is cooked through, 3 to 5 minutes. Squeeze the lemon over the lobster.
Transfer the pasta to serving bowls, cover with a spoonful of sauce and top with the cooked lobster. Garnish with tarragon leaves and serve.