Halve or coarsely chop the mushrooms. In a medium saucepan, melt the butter. Add the mushrooms and the shallots and cook until all of the liquid has cooked out of the mushrooms and the shallots are tender. Add the port wine; boil the mixture gently for 1 minute.
Stir in the whipping cream and thyme; bring to boiling. Boil gently for 10 minutes or until reduced to 1-1/4 cups. Stir in the capers. Season with salt and pepper. Remove from heat; keep warm.
Rinse fillets. Pat dry with paper towels. Season fillets with salt and pepper; coat with flour.
In a large skillet, heat canola oil. Place fish in hot oil. Cook until fish is golden brown and flakes easily with a fork, carefully turning once. Allow 3 to 4 minutes per side for 1/2-inch-thick fillets and 5 to 6 minutes per side for 1-inch-thick fillets.
Transfer the fish to warm serving plates; ladle the mushroom sauce over the fish.
When preparing the sauce, you can substitute dried morel mushrooms for the fresh morels, if you like. Rehydrate the mushrooms according to package directions and drain off the liquid.