Purée the raspberries with the icing sugar and lime juice, cover and chill.
Bring the water, butter, vanilla seeds and salt to the boil, reduce the heat. Combine the flour, psyllium and baking powder, add all at once, stir with a risotto spoon for approx. 1 1/2 mins. until a smooth dough is formed which separates from the base of the pan. Remove the pan from the heat.
To mix in the eggs
Using the whisk attachment on a hand mixer, gradually whisk in the eggs; the dough should be soft but not runny. Transfer the dough to a piping bag with a serrated nozzle (approx. 10 mm in diameter)
To deep-fry the churros
Fill a pot to 1/3 with oil and heat it to approx. 160°C. Pipe the dough (approx. 8 cm long) into the hot oil in batches, cut with scissors, deep-fry for approx. 5 mins. until golden brown. Remove and drain on paper towels. Mix the sugar and cinnamon, toss the churros in the sugar.