Place the potatoes into a large pot and cover with water. Bring to a boil over high heat; reduce heat to medium-low, cover and simmer until tender, about 20 minutes.
While the potatoes are boiling mash the garlic with 1/2 teaspoon of salt into a smooth paste. Drain potatoes then mash together with the garlic paste, butter and pepper. Keep warm.
Preheat an oven to 260 degrees C.
Pulse the hazelnuts and bread crumbs in a food processor until finely ground but not ground into a paste. Pour onto a shallow dish. Season the fish fillets on both sides with 1/4 teaspoon of salt. Gently press one side of the fillets into the nut mixture and set aside.
Heat the olive oil and 1 tablespoon of butter in an ovenproof frypan over medium-high heat. Place the fillets nut-side-up into the frypan; cook until the fish begins to brown, 3 to 4 minutes. Turn the fillets over and place into the preheated oven. Bake nut-side-down until the fish flakes easily with a fork, about 5 minutes.
Melt the remaining 5 tablespoons of butter in a small saucepan over medium-high heat until it turns nut brown. Stir in the vegetable stock and bring to a simmer.
To serve, mound the potatoes onto the center of each dinner plate. Place a fillet on top of the mashed potatoes nut-side-up then drizzle with the browned butter sauce.