Make wild-rice risotto:.
Bring water with salt to a boil in a heavy saucepan, then simmer wild rice, covered, until tender, about 45 minutes. Remove from heat and let rice stand in cooking liquid, uncovered, 30 minutes. Drain rice in a sieve set over a saucepan and reserve rice. Add vegetable stock to cooking liquid and bring to a simmer. Keep liquid hot, covered, over low heat.
Heat butter and oil in a 6- to 8-quart heavy pot over moderately high heat until hot but not smoking, then sauté sage, stirring, 1 minute. Add onions and sauté, stirring, until softened. Stir in celery with salt to taste and sauté, stirring, until onions begin to turn golden brown, about 7 minutes.
Stir in Arborio rice and cook over moderate heat, stirring constantly, 2 minutes. Add wine, thyme, and pepper to taste and cook, stirring constantly, until wine is absorbed. Stir in 1 cup hot broth and simmer, stirring frequently, until broth is absorbed. Continue adding broth, 1 cup at a time, and simmering, stirring frequently and letting each addition be absorbed before adding the next, until rice is tender but still al dente and creamy-looking and liquid is absorbed, 20 to 22 minutes. Toward end of cooking, stir in wild rice.
Remove pot from heat and stir in Parmesan, parsley, and salt and pepper to taste. Cool risotto and chill, covered, at least 4 hours and up to 12. Bring risotto to room temperature before proceeding with recipe.
Make risotto cakes:.
Lightly beat yolks and stir into risotto. Beat whites and a pinch of salt with an electric mixer until they just hold stiff peaks. Stir half of whites into risotto to lighten slightly and fold in remaining whites. Let batter stand at room temperature 15 minutes before proceeding.
Heat 1 tablespoon oil in a 12-inch nonstick skillet over moderate heat until hot but not smoking. Scoop 4 (rounded 1/4-cup) mounds of batter into skillet, flattening with back of a spoon to form cakes 3 inches across by 1/2 inch thick. Cook cakes until golden brown on underside (this is critical to prevent cakes from breaking when turning), adjusting heat if browning too quickly, 5 minutes. Carefully turn cakes over with a metal spatula and cook until golden brown on underside, 3 to 4 minutes. Transfer as browned to a large shallow baking pan and arrange in one layer. Make more cakes in same manner, wiping skillet clean with paper towels between batches and adding more oil as necessary.
Preheat oven to 450°F.
Just before serving, bake risotto cakes until sizzling, about 6 minutes. Serve immediately.