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Mash 3 tablespoons of softened butter with the mustard in a ramekin. Mix well with a fork and reserve.
Prepare the kidney. Keeping the kidney whole, peel off any filament and using a paring knife, cut away the knob of fat in the middle. As you move around the kidney removing the fat, take away any obvious connective tissue as well. If you don't get all the fat or connective tissue, it's ok. You don't want to pare away too much (or any) of the kidney itself either.
Heat 4 tablespoons of butter in a pan (that will snugly hold your kidney whole) over medium-high heat. Wait while the butter foams. Then, as the foaming subsides, add the kidney whole to the pan. Cook for about 10 minutes, turning every couple of minutes. Keep the heat steady so that the kidney cooks, but the butter does not brown. If your kidney fits your pan well, this is easier to do; if there is extra space in the pan, you may need to lower the heat a little to keep the butter from burning. The kidney may brown a little, but should not cook through. When all the sides are cooked and the kidney is still bouncy to poke and pink in the middle, remove it to a plate.
While keeping the cooked kidney juices and butter in the pan, add the minced shallots. Cook for a minute or two until soft, then add the wine and lemon juice together. Turn up the heat for a moment and bring to a boil. Reduce the heat a bit, but continue to boil off the liquid until it has reduced to about half.
Remove the pan from the heat. Add your reserved mustard butter, a spoonful at a time - stirring it into your sauce - until it is all combined. Season your sauce with salt and pepper.
Return to your reserved kidney. Slice the kidney crosswise and season with salt and pepper. Return these seasoned slices and their juices to the pan with your sauce. Add parsley to the pan and mix well for just a minute over low heat to warm the kidney through (without cooking any further!). When slices are evenly covered in sauce, serve immediately!