For the pickled cucumber, peel and seed the cucumber and slice into 1cm/½in pieces. In a bowl, whisk together the mustard, salt, sugar and vinegar. Marinate the cucumber slices in the mixture for at least 15 minutes and up to one hour.
For the duck, check the duck breasts and trim off any fat and sinew. Season with salt and freshly ground black pepper.
Heat a large heavy frying pan over a high heat and sauté the duck breasts in the vegetable oil over a medium to high heat until well browned but still pink in the middle. This will take about four minutes on each side.
Remove the duck breasts. Add all the teriyaki glaze ingredients to the pan and boil the mixture until it becomes thick and syrupy. Add the duck to the glaze, and turn frequently for about one minute. When well glazed on both sides, place the breasts on a cutting board and slice into 4-5 pieces each.
Meanwhile, for the celery tempura, heat the oil in a deep, heavy-bottomed pan. It is ready when a breadcrumb that is dropped in will sizzle gently in it. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
In a small bowl, beat the egg yolk with the water until just mixed. Add the flour and stir just five times with a fork and no more. The mixture will be very lumpy, and this is part of the secret to good tempura.
Dip the celery leaves in the dusting flour, and then into the tempura batter. Fry in the hot oil for about three minutes until golden brown, turning the leaves in the oil to ensure even cooking. Remove with a slotted spoon and season lightly with salt.
Serve the duck with the pickled cucumber and garnish with the celery leaf tempura.