For the ravioli: Heat the oil in a medium saucepot over medium heat. Add the shallots and sweat until translucent, about 2 minutes. Add the garlic and sweat until fragrant, about 30 seconds. Deglaze with the white wine. Bring to a gentle simmer and add 1/2 stick of the butter. Stir until melted. Add the basil, parsley and tarragon. Add the lobster pieces and poach in the liquid until almost cooked through, 1 1/2 to 2 minutes. Stir in the lemon zest and lemon juice. Remove from the heat, transfer the lobster to a dish and set aside to cool. Reserve the liquid.
Bring a medium saute pan to medium-low heat. Add the flour and toast gently for about a minute. Add the salt and pepper and the remaining 1/2 stick butter, stirring constantly. Cook until the starch begins to dissipate, about 3 minutes; be careful to not allow the roux to darken. Whisk the reserved poaching liquid into the roux until smooth. Simmer until the starch is completely cooked out and the sauce is thickened, 7 to 10 minutes. Remove from the heat, transfer to a medium mixing bowl and cool completely.
Gently fold the lobster into the thickened poaching liquid. Lay out the wonton wrappers, working in small batches so they don't dry out. Spoon 1 tablespoon of the filling into the center of each, brush the edges with water to moisten, place a second wonton over top and press to adhere. Use an inverted 2-inch cookie cutter (the smooth side) to gently seal the filling into the middle (this ensures a snug fit and no air bubbles which will make the ravioli burst when poached). Use a 3-inch cookie cutter (the sharp side) to cut out and create round raviolis.
Bring 2 quarts well-salted water to a simmer in a large pot. Working in batches, simmer the ravioli until they are al dente, 3 to 4 minutes. Remove with a slotted spoon and set aside.
Meanwhile, make the brown butter sage sauce: Set a large skillet over medium heat. Add the butter and cook, stirring until it melts and becomes foamy. Once the butter turns golden brown and has a nutty aroma, about 3 minutes, add the capers, sage and garlic. Season lightly with salt and pepper. Once the sage is fragrant, remove the skillet from the heat and fold in the parsley, tarragon and lemon juice. Stir to incorporate and warm up the herbs and lemon -- this will release their aroma but they'll maintain their bright vibrant color.
To heat and serve: Working in batches to not overcrowd the pan, gently toss the poached ravioli in the brown butter in a large saute pan for 1 to 2 minutes over medium heat to coat the pasta evenly. Place the ravioli on plates and spoon over sauce from the pan. Top with shaved Parmesan, garnish with basil buds and a drizzle of extra-virgin olive oil.