Step 1
Whisk together the flour and ¼ teaspoon each salt and pepper in a shallow plate and coat the fish on one side only. Shake off excess flour and transfer fish flour-side up to a sheet pan or cutting board. Reserve 1-1/2 teaspoons of the flour mixture for the sauce.
Step 2
Heat 1 tablespoon of the oil in a large stainless steel or well-seasoned cast iron skillet over high heat until just smoking. Add as much sole as will fit comfortably in a single layer, flour-side down and cook until bottom is browned, about 2 minutes. With two spatulas, turn the fish and remove pan from the heat. Let stand until just cooked through, about 30 seconds to 1 minute longer. Remove fish to a platter, cover to keep warm and wipe out the skillet. Repeat with the remaining oil and fish. Wipe out the skillet to make the sauce.
Step 3
Melt the butter in the same skillet over medium heat and add the shallots. Cook until softened, about 2 minutes, and whisk in the 1-1/2 teaspoons of the reserved flour. Cook for 30 seconds.
Step 4
Whisk in the wine, water, mustard, lemon zest and half the lemon juice and cook for 2 minutes. Season with salt and pepper, add more lemon juice if needed. Spoon sauce over the fish. Garnish with chopped parsley.
