I only need to read the name of this sauce and I see the fillets of sole in the hot sauté pan, delicate and golden around the edges, the perfect sweet partner for this zesty but elegant sauce. Or, to make a slightly lighter dish, I might cook the sole en papillote — seasoned with salt and pepper, wrapped in parchment (or aluminum foil, let’s make things easy), and baked with no added fat. Bake the packets on a large rimmed baking sheet in a 400-degree oven for five to eight minutes, depending on the thickness of the fish. It’s so simple and delicious. The sauce is also excellent on braised or grilled leeks, or as a dipping sauce for steamed artichoke.
Storage: To keep the sauce warm, hold it in the saucepan for a few minutes. For longer keeping, transfer it to a bowl set over a saucepan of hot water (the water should be no hotter than 110 degrees). You can also keep the sauce in a thermos for up to 2 hours. The sauce cannot be stored in the refrigerator, as it will separate when reheated.
In a small saucepan, combine the lemon juice, vinegar, and shallot and bring to a simmer over medium-high heat. Cook, stirring and scraping the sides of the pan occasionally with a heat-resistant rubber spatula, until the liquid is reduced to about 2 tablespoons, 4-5 minutes.
Add the butter and start stirring it around in circles with a whisk, a fork, a wooden spoon, or anything that lets you "stab" the hunk of butter so that you can move it around easily. The butter will start to melt and the mixture will bubble and boil around the edges. The melted butter should look creamy, rather than melted and oily. Keep stirring and blending until almost all of the butter is incorporated, then remove the pan from the heat as you work in the last bit. Whisk in the capers, parsley, lemon zest, ¼ teaspoon salt, and a few generous grinds of pepper. Taste and adjust the seasoning with salt and pepper if needed. If the sauce is still very sharp, whisk in another ½ tablespoon cold butter. If possible, serve right away.
Variation: Omit the parsley and add 1 tbsp chopped fresh dill and 2 tsp finely diced cornichon with the capers.