In a 12-inch skillet over medium-high heat, in hot olive oil, sauté mushrooms, minced tarragon, pepper, and 1/2 teaspoon salt until mushrooms are golden brown, about 10 minutes.
Add cubed turkey and 1/3 cup water, stirring to loosen all browned bits from the bottom of the skillet. Keep warm.
Meanwhile, in a 3-quart saucepan, place 1/2 teaspoon salt and 1-1/3 cups milk; gradually whisk in cornmeal until smooth. In a 2-quart saucepan, heat remaining 2/3 cup milk and 2 cups water to boiling over high heat; whisk into cornmeal mixture. Heat to boiling over medium-high heat, whisking. Reduce heat to low. Cook while stirring, 5 minutes or until thick. Stir in Parmesan cheese.
Serve polenta with turkey/mushroom mixture. Garnish each plate with a sprig of fresh tarragon leaves.