For the dough, warm the milk to 45C/113F on a cook’s thermometer. Place the salt, sugar, eggs, lime zest, flour and yeast in the bowl of a freestanding mixer with a dough hook attachment. Add 130ml/4fl oz water.
Pour in two-thirds of the milk and begin to mix on a low speed using a dough hook. Gradually add more milk until all the flour has been incorporated into a dough, you may not need all the milk.
Increase the speed to medium and mix for about three minutes. Slowly add the butter and mix for another five minutes, or until the dough is smooth and elastic. Place in a large bowl and cover with cling film. Set aside to prove in a warm place (ideally 27C/80F) for 45 minutes.
Meanwhile, make the raspberry and mint jam. In a food processor blend the raspberries, mint leaves and lemon juice to a smooth purée.
Pass through a sieve and weigh how much purée you have. Place the purée in a large saucepan and add the same weight in jam sugar (you may not need all 300g/10½oz of jam sugar). Heat gently until the sugar is melted.
Add a knob of butter and increase the heat to bring the mix to a rolling boil. Boil for four minutes then pour the jam into a bowl to cool and set.
For the doughnuts, tip the dough out onto a lightly floured work surface. Knock the air out gently and stretch and fold the dough back onto itself a couple of times.
Roll the dough out to around 1cm/½in thick and cut out the donuts using a plain round 9cm/3½in cutter for the outer ring and a 3cm/1¼in cutter to cut out the centre.
Grease two large baking trays with butter. Place the circles of dough on the trays, leaving room for expansion in between. Spray lightly with sunflower oil and prove for 20 minutes in a warm place.
Preheat a deep-fat fryer to 180C/350F, or alternatively heat a large, heavy-based pan of vegetable oil until a piece of bread sizzles and turns golden-brown when added to the oil. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
Fry the donuts in small batches, turning frequently, until golden-brown. It will take approximately three minutes per batch.
Place the cooked doughnuts on wire racks covered with kitchen roll and leave to cool.
When cool, fill the donuts with the raspberry and mint Jam.
Spoon the jam into a piping bag fitting with a filling nozzle and squeeze jam into each ring doughnut at four equally spaced points around the edge.
For the white rum glaze, add all the ingredients to a saucepan and bring to the boil. Cook until the sugar is dissolved then set aside to cool slightly.
Place the doughnuts directly on wire racks set over kitchen roll (to catch any drips). Brush the glaze over each doughnut until evenly covered.
For the decoration, cut the raspberries in half and place four halves on the top of each doughnut. Set aside to allow the glaze to cool and set.
When the glaze is set, finish decorating the doughnuts with the mint leaves.