Preheat the oven to 350F.
Combine the Parmesan, bread crumbs, and butter until thoroughly blended, and press the mixture into the bottom of a 9-inch springform pan.
Heat the oil in a large skillet over high heat. Add the onions and the green and red bell peppers and sauté, stirring and shaking the skillet, for 2 minutes. Stir in the mushrooms and the salt and pepper, sauté for 1 to 2 minutes, and remove from the heat.
Using an electric mixer, beat the cream cheese with the eggs in a large bowl until very thick and frothy, for about 5 minutes. Beat in the cream, Gouda, sautéed vegetables, and crabmeat. 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. If you refrigerate the cake, allow it to come to room temperature, for about 1 hour before serving.
While the cheesecake is cooling, prepare the Green Onion Coulis.
To serve, cut the cake in wedges with a warm knife and serve each wedge with about 3 tablespoons of the coulis.
Makes 12 to 16 brunch servings.