Combine the water, wine, carrot, celery, onion, lemon slices, bay leaves, peppercorns, parsley stems, and fine sea salt in a 30 cm / 12-inch deep sauté pan. Bring to a boil, then lower the heat and simmer partly covered for 10 minutes, until the broth smells of lemon and bay.
While the broth simmers, put the potatoes in a 3-quart / 2.8 L saucepan, cover with cold water by 3 cm, and add the coarse salt. Bring to a boil and cook 12 minutes, add the green beans, and cook 4 minutes more, until a knife slides into the potatoes and the beans are bright green. Drain.
Pat the fillets dry and slide them into the broth in a single layer. The liquid should reach the top of the fish, so add hot water if needed. Turn the heat to its lowest setting so the surface barely trembles, about 85 C / 185 F, cover, and poach for 7 minutes. Don't let it boil: hake is lean and soft, and a rolling boil shreds it. The fish is done when it's opaque white throughout and flakes under a fork, at 145 F / 63 C. If the center looks glassy, cook 1 minute more.
Whisk the olive oil, lemon juice, 3 tbsp warm poaching broth, parsley, capers if using, and a pinch of flaky salt in a small bowl until loose and slightly cloudy. Lift the fillets out with a fish spatula and drain on a folded towel for 10 seconds. Plate with the potatoes and beans, spoon over the dressing, and finish with flaky salt. Serve within 10 minutes.