Place the hazelnuts on a baking sheet in a preheated 350-degree oven for 10 minutes. When they are lightly toasted, remove from the oven and immediately rub them vigorously in a heavy, dry kitchen towel to remove the skins. Set aside to cool.
Place the new potatoes, the minced shallots, and the chicken stock in a 2 quart sauce pot. Add 1 teaspoon of salt, and bring to a boil. Lower the heat to a simmer and cover with a tight-fitting lid. Cook for 20 minutes, or until the potatoes are done, but firm. The tip of a knife should penetrate them with a slight resistance.
Meanwhile, cut off the head and tail of the trout. Using tweezers, remove any pin bones you may find in the trout. Separate the trout into 2 fillets each by cutting down the back of the trout. Season with salt and pepper, and dredge the trout fillets in the flour.
Add the green beans to the potato pot, cover and place on medium-high heat for 5 minutes.
Preheat a 12-inch skillet over medium heat. Add the olive oil and 2 tablespoons of butter. Saute the trout fillets, skin side down, until golden brown. Turn the fillets and cook 1 more minute. Remove from the pan, and repeat until all the trout are cooked. Place all the fillets a baking sheet in the oven to keep warm. Add the contents of the potato pot to the trout pan. Add the hazelnuts, bring to a boil, and stir in the lemon juice and butter.
To serve, divide the potato mixture onto 4 plates, and place 2 fillets, skin side up, crisscrossed on each plate. Spoon the remaining hazelnut sauce over each portion, and serve immediately.