In a large bowl, combine the flour, sugar, salt, and active dry yeast.
Pour the warm milk into the dry ingredients, then add the egg, melted butter, and vanilla extract.
Mix the ingredients until combined, then knead the dough for 8-10 minutes, until it becomes smooth and elastic.
Cover the dough and let it rise in a warm place for about 1 hour, or until doubled in size.
Meanwhile, prepare the lobster filling: mix the cooked, chopped lobster meat with mayonnaise, lemon juice, chives, and ground black pepper.
On a lightly floured surface, roll out the dough to about 1 cm thickness. Use a glass or doughnut cutter to cut out circles.
Place a small amount of lobster filling in the center of each dough circle, then fold it in half and seal the edges well to prevent the filling from escaping during frying.
Place the filled doughnuts on a baking sheet lined with parchment paper, and cover them with a clean kitchen towel. Let them rise for an additional 30 minutes.
In a large pot, heat the oil to approximately 340°F (170°C). Test the oil by dropping in a doughnut: if it immediately rises to the surface and gently bubbles, the oil is ready.
Fry the filled doughnuts for 2-3 minutes per side, until golden brown. Use a slotted spoon to remove them, then place them on a paper towel to absorb excess oil.
Sprinkle the tops of the still-warm doughnuts with powdered sugar to create a perfect balance of sweet and savory flavors.
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