Remove the transparent membrane covering the kidneys. Cut the kidneys in half crosswise and remove the core from the middle.
Melt the butter in a non-stick frying pan. Put the kidneys in the pan, cut side down. Cook over a moderately high heat for 2 minutes, then turn and cook for a further 2 minutes, pressing them flat with a spatula as they cook. Take care not to overcook the kidneys or they will be dry and rubbery. Remove the kidneys to a plate.
Pour the vinegar into the pan and scrape up any meaty sediment using the spatula. Cook over a high heat until most of the vinegar has evaporated. Add the mustard and cream and bring almost to boiling point, stirring all the time. Stir in the tarragon and season to taste.
Return the kidneys to the pan, reheat them for a few seconds, then serve immediately with the creamy sauce. Garnish with fresh tarragon.