Peel the carrot, onion, and shallot. Slice the carrot and finely chop the onion. Sauté them in a large pot, deglaze with half of the white wine, add 17 fl oz of water and the bouquet garni. Cover and simmer for 30 minutes. Preheat the oven to 392°F (th.6-7).
Strain the broth, return it to the heat, and cook the peas and broad beans in it for 5 minutes. Drain them and then plunge them into ice water.
Slice 1 lemon into thin rounds, wash and dry the dill. Line a large ovenproof dish with parchment paper and lay the lemon slices at the bottom. Place the hake fillets on top, season with salt and pepper, add some dill sprigs, and cover with the remaining lemon slices. Fold the parchment paper to make a papillote and bake for 20 minutes.
Zest 1 lemon and juice it. Finely chop the shallot and sauté it with the butter, deglaze with the remaining white wine, add the mustard and cream, mix well, and let it simmer over low heat.
Sauté the pre-cooked broad beans and peas in a pan with a drizzle of olive oil, season with salt and pepper, and deglaze with the lemon juice.
Remove the dill and lemon from the dish to retrieve the hake fillets. Cut the last lemon into quarters. On each plate, pour some sauce, place a hake fillet and the vegetables, garnish with lemon zest, dill sprigs, a lemon quarter, sprinkle with crushed pink peppercorns, and serve immediately.