1. For the new potatoes, bring a large amount of water to a boil in a pot. Cut the larger potatoes in half if necessary and cook them for 12 to 15 minutes. Trim the green beans. Pour a little water into another pot with a pinch of salt, add the green beans. Bring to a boil, covered, for 8 to 10 minutes. Drain both and set aside without a lid.
2. Cut the lemon in half lengthwise. Slice one half and cut the other half into quarters. Mince the shallot. Finely chop the dill. Mince or crush the garlic.
3. In a small pot, gently bring a small amount of water to a boil mixed with the fish stock cube, lemon slices, half of the dill, and half of the shallot. The water should be enough to cover the salmon fillet. Place the salmon fillets in the pot when the water is simmering, then cover. Cook them for 2 minutes, then remove from heat and let them cook for an additional 5 minutes, still covered.
4. Carefully remove the salmon from the pot, place it on a sheet of aluminum foil, and close it. Pour the broth into a measuring cup and remove the lemon slices. Pour 2 tablespoons of broth per person back into the small pot and gently bring to a boil. Incorporate the heavy cream into the broth and heat for 1 to 2 minutes. Season with salt and pepper.
5. Mix the garlic with the green beans. Mix the remaining shallot and 1 teaspoon of mustard per person with the new potatoes.
6. Serve the poached salmon on plates. Garnish with the sauce, capers, remaining dill, and lemon quarters. Serve with the green beans and new potatoes.