Line a large baking sheet with nonstick baking paper. Roll the pastry out on a lightly floured surface to 5 mm (¼ inch) thick. Stamp out 40 x 6 cm (2½ inch) rounds using a biscuit or pastry cutter. Put 20 of the rounds on the baking sheet, spaced well apart, and brush with beaten egg. Using a 3 cm (1¼ inch) cutter, stamp out circles from the centre of the remaining rounds. Discard the inner pastry circles, leaving you with 20 pastry 'rings'. Put these 'rings' on the brushed pastry rounds and press gently to seal. Brush again with the beaten egg, then bake in a preheated oven, 200°C (400°F), Gas Mark 6, for 12–15 minutes, or until risen and golden. Remove from the oven and transfer to a wire rack to cool completely.
Meanwhile, put the lobster meat in a bowl and mix in the mayonnaise, mustard, red pepper and chopped tarragon. Season well with salt and pepper. Using a teaspoon, carefully spoon the mixture into the cold puff shells. Garnish with tarragon sprigs and serve immediately.