Preheat the oven to 400 degrees F.
Combine the pecans and bread crumbs in a food processor. Process until they are coarse crumbs then transfer to a shallow dish.
Place the flour in a shallow dish. Place the beaten egg in another shallow dish.
Season the grouper with salt and pepper. Dredge each piece first in the flour, then in the egg, letting any excess drip off. Press each side of the fish into the pecan crumbs to coat.
Melt 1/4 cup of the butter in an oven-proof skillet over medium-high heat. Cook the fish for one minute on each side. Place the skillet in the oven and bake for 10 minutes or until the fish flakes easily with a fork. Transfer the fish to a serving platter and keep warm.
Wipe out the skillet and add the remaining butter to it. Cook over high heat until the butter starts to brown, stirring constantly. Stir in the lemon juice and parsley. Pour the browned butter sauce over the fish and serve immediately.