Preheat the oven to 350 degrees F. Combine the Parmesan, bread crumbs and butter until thoroughly blended, and press into a 9-inch springform pan. In a saute pan, heat the olive oil. Add the onions, shallots, garlic and peppers, and saute for 2 minutes. Stir in the mushrooms and season with salt and pepper, saute for 1 to 2 minutes, and remove from heat. Using an electric mixer beat the cream cheese with the eggs in a large bowl until very thick, about 5 minutes. Beat in the cream, Gouda, sauteed vegetables and crab meat. Mix until thoroughly incorporated and creamy, for about 2 minutes. Pour the filling over the crust in the springform pan and bake until firm, for about 1 hour. Remove from the oven and allow to cool to room temperature.
For the sauce: In a small mixing bowl, combine all the ingredients together, mix to incorporate.
Season with salt and pepper. Chill. Spoon a pool of the coulis in the center of a plate. Place a piece of the cheesecake in the center of the coulis. Garnish with the egg yolks, egg whites, chives, long chives, capers and red onions.