Pat the capitone pieces dry with paper towels. Toss them in a bowl with the lemon juice, fine sea salt, and pepper, then cover and refrigerate for 15 minutes.
Drain the pieces and pat them dry again, pressing until no moisture shows on the skin. Wet eel spits hard in hot oil and turns the flour coat pasty, so don't skip this.
Heat the oven to 90 C / 200 F and set a wire rack over a 33 x 46 cm / 13 x 18-inch sheet pan. Pour the oil into a 24 cm / 9.5-inch Dutch oven, about 2.5 cm / 1 inch deep, and heat over medium heat to 170 C / 340 F, about 6 minutes.
Spread the flour in a shallow dish. Dredge 7 pieces just before frying, turn to coat thinly, and shake off the excess.
Lower the pieces into the oil, placing them away from you. Fry for 8 to 10 minutes, turning once halfway, until deep golden and a thermometer in the thickest piece reads 145 F / 63 C. Lift onto the rack, sprinkle with flaky salt, and move to the oven.
Let the oil return to 170 C / 340 F, about 2 minutes, then dredge and fry the second batch the same way. Serve within 20 minutes with lemon wedges and parsley.