To make the pasta dough, combine the flour, eggs, and salt in the bowl of a stand mixer fitted with the dough hook. Mix on low speed until a shaggy dough forms, then increase to medium and knead for 5 minutes. Transfer the dough to a lightly floured surface and knead by hand for 2 minutes until smooth. Wrap in plastic and let it rest at room temperature for at least 30 minutes.
While the dough rests, make the filling. Heat olive oil in a saucepan over medium heat. Add the shallot and cook until softened, about 2 minutes. Stir in the garlic for 30 seconds. Pour in the white wine to deglaze the pan, scraping up any browned bits. Simmer for 2 minutes.
Reduce heat to low and stir in 4 tablespoons of butter until melted. Add the tarragon, basil, and parsley. Add the lobster pieces and gently poach until the meat is firm and cooked through, about 2-3 minutes. The internal temperature should reach 145°F (63°C). Use a slotted spoon to transfer the lobster to a bowl, reserving the poaching liquid in the pan. Let the lobster cool slightly.
Transfer the cooked lobster to a food processor and pulse a few times until coarsely chopped. Do not over-process into a paste. Return the lobster to a bowl and stir in the lemon juice. Season with salt and pepper to taste. Set aside.
Cut the rested pasta dough into four equal pieces. Working with one piece at a time (keep others covered), flatten it and run it through a pasta machine on the widest setting. Fold and pass it through the widest setting again. Continue passing the dough through progressively narrower settings until you have a long, thin sheet (about 1/16-inch thick).
Lay one pasta sheet on a lightly floured surface. Place heaping teaspoons of the lobster filling about 2 inches apart down the center of the sheet. Lightly brush water around the filling. Place a second pasta sheet on top. Press firmly around each mound of filling to seal the pasta and remove any air pockets. Use a pastry wheel or knife to cut into individual ravioli.
Bring a large pot of salted water to a rolling boil. While the water heats, make the sauce. Melt the remaining ½ cup of butter in a large skillet over medium heat. Add the sage leaves and cook, swirling the pan occasionally, until the butter browns and smells nutty, about 4-5 minutes.
Carefully add the ravioli to the boiling water and cook until they float to the surface, about 3-4 minutes. Stir 2 tablespoons of the reserved lobster poaching liquid into the brown butter sauce.
Using a slotted spoon, transfer the cooked ravioli directly from the pot into the skillet with the brown butter sauce. Gently toss to coat. Serve immediately, garnished with extra fresh parsley if desired.