Heat a large skillet over medium flame for 2-3 minutes. Season the monkfish generously with salt and pepper. Add the olive oil to the pan and heat 20-30 seconds. Place the seasoned fish in to pan evenly and cook for 1 minute. Turn the flame down to low. Turn the fish periodically and gently brown all of the sides of the fish. Turn the flame off and remove the fish to a platter with parchment paper or other.
Preheat oven to 450 degrees. In a small sauce pot, heat the water, ½ of the extra-virgin olive oil, a pinch of salt and 1 bay leaf to a boil. Add the carrots and cook 2-3 minutes. Remove the carrots from the heat and strain.
Place a sheet of foil covered by a sheet of parchment paper on a countertop. Place ¼ of the cooked carrots, the olives, the tomatoes, 1 sprig of fresh thyme and, 1 garlic clove just left of the center of the foil and paper sheets. Place a piece of cooked fish on top of this pile. Spoon ¼ of the butter, the remaining extra-virgin olive oil, and an ¼ of the wine over the fish. Repeat with the remaining fillets. Fold the right side of the paper and foil sheets evenly over the left side and start folding the edge of the foil from one open end around to the other open end, making a well-sealed package. Carefully make small folds and the package will seal. Place the packages on a cookie sheet and bake in the pre-heated oven for 8-10 minutes. Remove the tray from the oven and allow the packages to sit for 2-3 minutes. Cut the packages open with scissors and serve.