Special equipment: 2-inch circle cutter
Generously butter a large shallow baking dish and set aside. In a large saucepan, bring the chicken stock to a boil. Slowly pour in the polenta and the salt, whisking constantly with a wire whisk. Reduce the heat to low and cook, stirring constantly with a wooden spoon until thickened and smooth, about 20 to 25 minutes. Fold in the corn, Parmesan, and parsley. Pour the mixture into the prepared baking dish. Cover and chill in the refrigerator at least 4 hours or overnight.
Preheat the grill to medium-high. With a 2-inch circle cutter, cut the chilled polenta into circles. Brush with olive oil and season with salt and pepper. Grill on both sides until golden brown. Arrange the polenta on a platter and spoon the vinaigrette over. Sprinkle with chopped parsley.
In a blender, blend the vinegar, shallot, and mustard until smooth. With the motor running, slowly add the walnut oil and 1/2 cup of the olive oil until emulsified. Season with salt and pepper, to taste. Pour into a medium bowl and set aside.
Heat the remaining 1/4 cup of oil in a large saute pan over high heat. Add the mushrooms and cook until golden brown. Season with salt and pepper, to taste. Remove the mushrooms from the heat and coarsely chop them. Fold the mushrooms into the vinaigrette. Spoon the vinaigrette over the grilled polenta.