Preheat the oven to 374°F (th.6-7). Bring a large pot of salted water to a boil and cook the trimmed green beans for 3 minutes. Drain the beans and plunge them into ice water to stop the cooking process. Drain again and set aside.
Mix the butter with 2 tablespoons of olive oil and coarsely chopped dill. Place the salmon on a baking sheet lined with parchment paper, season with salt and pepper, and spread the butter mixture over the fish. Bake for 15 to 20 minutes.
While the fish is cooking, prepare the sauce. Set aside 1 lemon, zest the other 4 and reserve the zest. With a sharp knife, remove all the white pith from the peeled lemons. Cut these peeled lemons into 0.2-inch thick slices. In a pan, heat 3 tablespoons of olive oil, place the lemon slices in the pan and sear them on both sides until caramelized and golden.
Deglaze the pan with white wine to loosen the caramelized bits, then transfer everything to a blender. Add the remaining olive oil, honey, salt, and pepper, and blend quickly to obtain a smooth emulsion. Incorporate 4 tablespoons of water into the finished sauce.
Peel and chop the garlic cloves, sauté them in a hot pan with a drizzle of olive oil, add the green beans, and sauté for a few minutes. Season with salt and pepper, and serve immediately with the salmon, sauce, and the remaining lemon cut into quarters.