In a large bowl, whisk together the flour, sugar, salt, and active dry yeast.
Pour the warm milk into the dry ingredients, then add the egg, melted butter, vanilla extract, and key lime juice.
Mix the ingredients until just 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.
On a lightly floured surface, roll out the dough to about 1 cm (approximately 1/2 inch) thickness. Use a glass or a doughnut cutter to cut out circles.
Place the cut-out doughnuts on a baking sheet lined with parchment paper, and cover them with a clean kitchen towel. Let them rise for another 30 minutes.
In a large pot, heat the oil to about 340°F (170°C). Drop a doughnut into the oil as a test: if it immediately floats to the surface and gently bubbles, the oil is ready.
Fry the doughnuts for 1-2 minutes per side, until golden brown. Use a slotted spoon to remove them, and place them on a paper towel-lined plate to absorb the excess oil.
Once the doughnuts have cooled slightly, prepare the key lime glaze: whisk together the powdered sugar with the key lime juice until smooth.
Dip the tops of the doughnuts into the glaze, then sprinkle them with grated lime zest.
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