Clean, gut and wash the rainbow trout. Place it in a buttered ovenproof dish, surround it with chopped carrot, celery, onion and herbs. Season with salt and pepper, pour in 3 1/2 tablespoons of melted butter and cook.
As soon as the chopped vegetables are soft, add red wine and roast drippings and bring to a boil. Continue cooking in a preheated 350°F oven for about 25 - 30 minutes, basting the fish with its cooking juices every once in a while. Remove from the oven, drain and arrange on a serving platter. Keep warm.
Peel the potatoes and cut into big pieces. Bring potatoes to a boil in a pot of salted water and cook until soft enough to be easily pierced with a fork. Drain, put them in a bowl and toss with the remaining melted butter.
Pass the trout's cooking juices and vegetables through a food mill on the finest setting and pour into a saucepan. In a separate bowl, combine 1 tablespoon butter with 2 tablespoons flour and whisk with a spoon.
Add to the pan juices a little at a time. Bring sauce to a boil, whisking, and remove from the heat after 2 minutes. Dice remaining butter and add a bit at a time, whisking constantly. Season with salt. Cover the fish with sauce and serve immediately, with boiled potatoes on the side.