Pour the milk into a saucepan and add the split vanilla pod. Slowly bring to a simmer and once it begins to steam, remove from the heat and pour into a measuring jug. If it has reduced, top up to 130ml with extra milk, then leave to cool until tepid.
Put the flour into the large bowl of a stand mixer. Stir through the sugar, salt and yeast. Using the hook attachment for your mixer, begin combining the dry ingredients, then add the vanilla-infused milk and eggs. Keep mixing for 10 mins until you have a smooth, elastic dough. Scoop small spoonfuls of the softened butter and gradually add to the dough. When all the butter has been added, continue mixing for a further 5-6 mins or until it is well incorporated and the dough is sticky and stretchy.
Lightly oil a large mixing bowl, put the dough in the bowl, cover and leave in a warm place for 1 hr or until doubled in size.
Divide the dough into 20 equal-sized pieces (if you want to weigh them, they should be around 45g each). Roll the dough into balls and place them, well spaced apart, on two lightly oiled baking sheets. Cover with lightly oiled baking parchment, or a light tea towel (if it’s too heavy, it will keep the dough from rising) and leave for a further 1 hr 30 mins or until doubled in size.
Fill a large heavy-based saucepan two-thirds full with oil and place over a low-medium heat, bringing the temperature up to 170C. Use a cooking thermometer to check, then carefully lift each doughnut and gently lower into the oil to fry in batches of about 5 at a time. Cook each batch for 5 mins, turning halfway, until the doughnuts are dark golden brown. Remove with a slotted spoon and transfer to a baking tray lined with kitchen paper. Leave to cool completely.
Spoon the raspberry jam into a piping bag without a nozzle and snip a very small hole in the end. Use a skewer to poke a hole in the side of each doughnut, then push the end of the piping bag into each hole and pipe in a little of the jam. Set aside.
For the icing, mix together the icing sugar and 4 tbsp water, then a drop or two of the food colouring to make it pale pink. Drop a dessert-spoonful of the icing on top of each doughnut, then use a palette knife to spread and even out the icing. Sprinkle each one with the freeze-dried raspberries, if you like. Will keep in an airtight container for up to two days.