Heat a medium-sized, heavy-bottom casserole for 2 minutes over medium heat. Add ½ of the olive oil and heat for 20-30 seconds. Add the diced fennel, leeks, and celery. Add a pinch of salt and the fresh garlic cloves. Sauté the vegetable mixture for 3-4 minutes and then lower the flame. Cook the vegetable mixture gently until the vegetables soften, approximately 20 minutes. Stir the vegetables every 5 minutes or so.
Season the trout fillets with salt and pepper and dredge in flour. Heat a large skillet over medium flame for 2-3 minutes. Add the remaining olive oil. Place the trout in the pan skin side down and sauté both sides until golden brown, about 3-4 minutes a side. Remove the trout to a paper towel and then to serving plates. Pour out the extra oil and add the wine to the hot pan. Let it boil a bit and add the butter. Spoon the butter and wine reduction from the hot pan onto the cooked trout. Spoon some of the cooked vegetables as a garnish.