About 15 minutes before cooking, let the duck breasts sit at room temperature.
Preheat barbecue to high (about 245 °C – 475 °F).
Pat duck breasts dry with paper towels and, if desired, remove the white membrane. Remove excess fat from breasts.
Using a knife, score the skin of the breast in a crosshatch, making sure not to cut into the flesh. Season with salt and pepper.
Turn off the burners on one side of the barbecue. Place breasts, skin side down, on the hot and oiled barbecue grill over the opened burners. Cook for 1 minute 30 seconds to 2 minutes. Rotate breasts 45 degrees to form a grid pattern. Continue cooking for 1 minute 30 seconds to 2 minutes.
Turn breasts over and cook for 2 minutes.
Move breasts to the side of the barbecue with the burners off for indirect cooking, skin side down. Close the lid and continue cooking for 8 to 10 minutes, until the internal temperature reaches 58 °C (137 °F) on a meat thermometer. The breasts should be moderately firm to the touch and still pink inside. Remove breasts from the barbecue and transfer to a plate. Cover loosely with aluminum foil and let sit for 5 to 10 minutes before cutting.
Meanwhile, in a small saucepan, mix vinegar with white wine, shallots and parsley. Season with pepper. Cook over medium-low heat until liquid has reduced. Remove from heat.
In a bain-marie, whisk egg yolks and water with an electric mixer until smooth.
Stir white wine mixture into egg yolk mixture. Gradually add the melted butter for the sauce, whisking constantly until the mixture thickens. Add tarragon. Season with salt and stir. Remove from heat and keep warm.
Season lobster tails with salt and pepper.
In a small bowl, mix melted butter with paprika.
Place lobster tails on the hot and oiled barbecue grill and brush with seasoned butter. Close the lid and cook for 2 to 3 minutes on each side.
Heat oil in a skillet over medium heat. Cook gnocchi and asparagus for 4 to 5 minutes, stirring regularly. Season with salt and pepper.
Divide the gnocchi between the plates. Garnish with slices of duck breasts and lobster tails. Top with Béarnaise sauce.
