Line a baking tray with baking paper and liberally sprinkle it with flour.
To make the doughnuts, place the flour, baking powder, bicarbonate of soda, salt and sugar in a large bowl and mix to combine.
Whisk together the butter, egg yolks, vanilla and buttermilk in a jug. Make a well in the dried ingredients, pour in the buttermilk mixture and stir to form a thick cake mixture. Add the blueberries and mix briefly to distribute. Cover the bowl and chill in the fridge for a couple of hours or until firm.
Generously flour a work surface and roll out the doughnut dough until about 15mm/½in thick. Using a 7.5cm/3in round cookie cutter, cut out as many doughnuts as possible. Use a 2.5cm/1in cookie cutter to cut out holes from the centres of the doughnuts (these will be fried to make doughnut holes). Gather the scraps of the dough and press back together. If the dough is still cool and easy to handle, you can re-roll immediately and cut out more doughnuts. If it is a little sticky, chill the dough until firmed up. Place the doughnuts (and doughnut holes) on the prepared tray. Chill in the fridge.
Using either a deep fat fryer, a deep saucepan or a deep frying pan, pour in enough oil so it is a couple of inches deep. Make sure any pan still has plenty of room as you do not want oil near the rim. (CAUTION: hot oil can be dangerous. Do not leave unattended.) Preheat the deep fat fryer to 160–170C or place the pan over a medium heat and bring to the same temperature. Add a few doughnuts at a time to the oil and fry until golden-brown, flipping once halfway through cooking. The doughnuts should take 2–3 minutes to fully cook. Line a wire rack with kitchen paper and use a slotted spoon to transfer the doughnuts from the oil to the rack.
To make the glaze, heat the blueberries and lemon juice in a small saucepan for a couple of minutes or until the blueberries are starting to break down. Scrape the berries and juice into a jug and add the basil, bourbon, salt and vanilla. Add a little of the sugar and use a stick blender to blend until smooth. Add the rest of the icing sugar and mix until thick but still pourable. Adjust the thickness with either lemon juice if too thick or icing sugar if too thin. Pour the glaze over the doughnuts and set aside until the glaze has set. As doughnuts are fried, they are best enjoyed as soon as possible.
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